Proofgold Address

Param explicit_Field^{explicit_Field} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Known f16ac.. : ∀ x0 x1 x2 . ∀ x3 x4 x5 x6 : ι → ι → ι . (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x7 x8 = x5 x7 x8) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x7 x8 = x6 x7 x8) ⟶ explicit_Field x0 x1 x2 x3 x4 ⟶ explicit_Field x0 x1 x2 x5 x6
Param ap^ap : ι → ι → ι
Definition decode_b^decode_b := λ x0 x1 . ap (ap x0 x1)
Param lam^Sigma : ι → (ι → ι) → ι
Param ordsucc^ordsucc : ι → ι
Param If_i^If_i : ο → ι → ι → ι
Param encode_b^encode_b : ι → CT2 ι
Definition pack_b_b_e_e^pack_b_b_e_e := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 x4 . lam 5 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) (encode_b x0 x1) (If_i (x5 = 2) (encode_b x0 x2) (If_i (x5 = 3) x3 x4))))
Known pack_b_b_e_e_1_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 x5 x6 = decode_b (ap (pack_b_b_e_e x0 x1 x2 x3 x4) 1) x5 x6
Known pack_b_b_e_e_2_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x5 x6 = decode_b (ap (pack_b_b_e_e x0 x1 x2 x3 x4) 2) x5 x6
Definition field0^{RealsStruct_carrier} := λ x0 . ap x0 0
Definition field1b^{RealsStruct_plus} := λ x0 . decode_b (ap x0 1)
Definition field2b^{RealsStruct_mult} := λ x0 . decode_b (ap x0 2)
Param decode_r^decode_r : ι → ι → ι → ο
Definition RealsStruct_leq^{RealsStruct_leq} := λ x0 . decode_r (ap x0 3)
Definition field4^{RealsStruct_zero} := λ x0 . ap x0 4
Definition RealsStruct_one^{RealsStruct_one} := λ x0 . ap x0 5
Definition Field_of_RealsStruct^{Field_of_RealsStruct} := λ x0 . pack_b_b_e_e (field0 x0) (field1b x0) (field2b x0) (field4 x0) (RealsStruct_one x0)
Param explicit_Field_minus^{explicit_Field_minus} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι
Definition Field_minus^Field_minus := λ x0 x1 . If_i (x1 ∈ ap x0 0) (explicit_Field_minus (ap x0 0) (ap x0 3) (ap x0 4) (decode_b (ap x0 1)) (decode_b (ap x0 2)) x1) 0
Known Field_of_RealsStruct_0^{Field_of_RealsStruct_0} : ∀ x0 . ap (Field_of_RealsStruct x0) 0 = field0 x0
Known Field_of_RealsStruct_1^{Field_of_RealsStruct_1} : ∀ x0 x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ap (ap (ap (Field_of_RealsStruct x0) 1) x1) x2 = field1b x0 x1 x2
Known Field_of_RealsStruct_2^{Field_of_RealsStruct_2} : ∀ x0 x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ap (ap (ap (Field_of_RealsStruct x0) 2) x1) x2 = field2b x0 x1 x2
Known Field_of_RealsStruct_3^{Field_of_RealsStruct_3} : ∀ x0 . ap (Field_of_RealsStruct x0) 3 = field4 x0
Known Field_of_RealsStruct_4^{Field_of_RealsStruct_4} : ∀ x0 . ap (Field_of_RealsStruct x0) 4 = RealsStruct_one x0
Param struct_b_b_e_e^{struct_b_b_e_e} : ι → ο
Known pack_struct_b_b_e_e_I : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 x4 ∈ x0) ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ struct_b_b_e_e (pack_b_b_e_e x0 x1 x2 x3 x4)
Param unpack_b_b_e_e_o^{unpack_b_b_e_e_o} : ι → (ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ο) → ο
Known unpack_b_b_e_e_o_eq^{unpack_b_b_e_e_o_eq} : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 : ι → ι → ι . (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ unpack_b_b_e_e_o (pack_b_b_e_e x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition e08de.. := λ x0 x1 x2 x3 . and (RealsStruct_leq x1 x2 x3) (x2 = x3 ⟶ ∀ x4 : ο . x4)
Param RealsStruct^RealsStruct : ι → ο
Param encode_r^encode_r : ι → (ι → ι → ο) → ι
Definition pack_b_b_r_e_e^{pack_b_b_r_e_e} := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 : ι → ι → ο . λ x4 x5 . lam 6 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) (encode_b x0 x1) (If_i (x6 = 2) (encode_b x0 x2) (If_i (x6 = 3) (encode_r x0 x3) (If_i (x6 = 4) x4 x5)))))
Known RealsStruct_eta^{RealsStruct_eta} : ∀ x0 . RealsStruct x0 ⟶ x0 = pack_b_b_r_e_e (field0 x0) (field1b x0) (field2b x0) (RealsStruct_leq x0) (field4 x0) (RealsStruct_one x0)
Param explicit_Reals^{explicit_Reals} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Known RealsStruct_explicit_Reals^{RealsStruct_explicit_Reals} : ∀ x0 . RealsStruct x0 ⟶ explicit_Reals (field0 x0) (field4 x0) (RealsStruct_one x0) (field1b x0) (field2b x0) (RealsStruct_leq x0)
Known RealsStruct_zero_In^{RealsStruct_zero_In} : ∀ x0 . RealsStruct x0 ⟶ field4 x0 ∈ field0 x0
Known RealsStruct_one_In^{RealsStruct_one_In} : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_one x0 ∈ field0 x0
Known RealsStruct_plus_clos^{RealsStruct_plus_clos} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field1b x0 x1 x2 ∈ field0 x0
Known RealsStruct_mult_clos^{RealsStruct_mult_clos} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field2b x0 x1 x2 ∈ field0 x0
Known RealsStruct_plus_assoc^{RealsStruct_plus_assoc} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field1b x0 x1 (field1b x0 x2 x3) = field1b x0 (field1b x0 x1 x2) x3
Known RealsStruct_plus_com^{RealsStruct_plus_com} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field1b x0 x1 x2 = field1b x0 x2 x1
Known RealsStruct_zero_L^{RealsStruct_zero_L} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field1b x0 (field4 x0) x1 = x1
Known RealsStruct_mult_assoc^{RealsStruct_mult_assoc} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field2b x0 x1 (field2b x0 x2 x3) = field2b x0 (field2b x0 x1 x2) x3
Known RealsStruct_mult_com^{RealsStruct_mult_com} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field2b x0 x1 x2 = field2b x0 x2 x1
Known RealsStruct_one_L^{RealsStruct_one_L} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field2b x0 (RealsStruct_one x0) x1 = x1
Known RealsStruct_distr_L^{RealsStruct_distr_L} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field2b x0 x1 (field1b x0 x2 x3) = field1b x0 (field2b x0 x1 x2) (field2b x0 x1 x3)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param explicit_OrderedField^{explicit_OrderedField} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Param lt^lt : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ι → ο
Param Sep^Sep : ι → (ι → ο) → ι
Param natOfOrderedField_p^{natOfOrderedField_p} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ο
Param setexp^setexp : ι → ι → ι
Known explicit_Reals_E^{explicit_Reals_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ο . (explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x7 ⟶ x5 x1 x8 ⟶ ∀ x9 : ο . (∀ x10 . and (x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) (x5 x8 (x4 x10 x7)) ⟶ x9) ⟶ x9) ⟶ (∀ x7 . x7 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ ∀ x8 . x8 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ (∀ x9 . x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (and (x5 (ap x7 x9) (ap x8 x9)) (x5 (ap x7 x9) (ap x7 (x3 x9 x2)))) (x5 (ap x8 (x3 x9 x2)) (ap x8 x9))) ⟶ ∀ x9 : ο . (∀ x10 . and (x10 ∈ x0) (∀ x11 . x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x7 x11) x10) (x5 x10 (ap x8 x11))) ⟶ x9) ⟶ x9) ⟶ x6) ⟶ explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ x6
Param iff^iff : ο → ο → ο
Known explicit_OrderedField_E^{explicit_OrderedField_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ο . (explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x5 x7 x8 ⟶ x5 x8 x9 ⟶ x5 x7 x9) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ iff (and (x5 x7 x8) (x5 x8 x7)) (x7 = x8)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ or (x5 x7 x8) (x5 x8 x7)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x5 x7 x8 ⟶ x5 (x3 x7 x9) (x3 x8 x9)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x1 x7 ⟶ x5 x1 x8 ⟶ x5 x1 (x4 x7 x8)) ⟶ x6) ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ x6
Theorem RealsStruct_leq_linear^{RealsStruct_leq_linear} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ or (RealsStruct_leq x0 x1 x2) (RealsStruct_leq x0 x2 x1) (proof)
Known RealsStruct_leq_plus^{RealsStruct_leq_plus} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ RealsStruct_leq x0 x1 x2 ⟶ RealsStruct_leq x0 (field1b x0 x1 x3) (field1b x0 x2 x3)
Definition RealsStruct_N^{RealsStruct_N} := λ x0 . Sep (field0 x0) (natOfOrderedField_p (field0 x0) (field4 x0) (RealsStruct_one x0) (field1b x0) (field2b x0) (RealsStruct_leq x0))
Known explicit_Field_of_RealsStruct^{explicit_Field_of_RealsStruct} : ∀ x0 . RealsStruct x0 ⟶ explicit_Field (field0 x0) (field4 x0) (RealsStruct_one x0) (field1b x0) (field2b x0)
Theorem explicit_OrderedField_of_RealsStruct^{explicit_OrderedField_of_RealsStruct} : ∀ x0 . RealsStruct x0 ⟶ explicit_OrderedField (field0 x0) (field4 x0) (RealsStruct_one x0) (field1b x0) (field2b x0) (RealsStruct_leq x0) (proof)
Param struct_b_b_r_e_e^{struct_b_b_r_e_e} : ι → ο
Param unpack_b_b_r_e_e_o^{unpack_b_b_r_e_e_o} : ι → (ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ι → ο) → ο
Definition OrderedFieldStruct^{struct_b_b_r_e_e_ordered_field} := λ x0 . and (struct_b_b_r_e_e x0) (unpack_b_b_r_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 : ι → ι → ο . λ x5 x6 . explicit_OrderedField x1 x5 x6 x2 x3 x4))
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known pack_struct_b_b_r_e_e_I^{pack_struct_b_b_r_e_e_I} : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 x4 ∈ x0) ⟶ ∀ x3 : ι → ι → ο . ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ struct_b_b_r_e_e (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5)
Known OrderedFieldStruct_unpack_eq^{OrderedFieldStruct_unpack_eq} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . unpack_b_b_r_e_e_o (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) (λ x7 . λ x8 x9 : ι → ι → ι . λ x10 : ι → ι → ο . λ x11 x12 . explicit_OrderedField x7 x11 x12 x8 x9 x10) = explicit_OrderedField x0 x4 x5 x1 x2 x3
Theorem RealsStruct_OrderedField^{RealsStruct_OrderedField} : ∀ x0 . RealsStruct x0 ⟶ OrderedFieldStruct x0 (proof)
Known tuple_5_1_eq^tuple_5_1_eq : ∀ x0 x1 x2 x3 x4 . ap (lam 5 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 (If_i (x6 = 2) x2 (If_i (x6 = 3) x3 x4))))) 1 = x1
Known tuple_6_1_eq^tuple_6_1_eq : ∀ x0 x1 x2 x3 x4 x5 . ap (lam 6 (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 (If_i (x7 = 2) x2 (If_i (x7 = 3) x3 (If_i (x7 = 4) x4 x5)))))) 1 = x1
Known encode_b_ext^encode_b_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ encode_b x0 x1 = encode_b x0 x2
Known decode_encode_b^{decode_encode_b} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ decode_b (encode_b x0 x1) x2 x3 = x1 x2 x3
Theorem Field_of_RealsStruct_1f^{Field_of_RealsStruct_1f} : ∀ x0 . RealsStruct x0 ⟶ (λ x2 . ap (ap (ap (Field_of_RealsStruct x0) 1) x2)) = field1b x0 (proof)
Known tuple_5_2_eq^tuple_5_2_eq : ∀ x0 x1 x2 x3 x4 . ap (lam 5 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 (If_i (x6 = 2) x2 (If_i (x6 = 3) x3 x4))))) 2 = x2
Known tuple_6_2_eq^tuple_6_2_eq : ∀ x0 x1 x2 x3 x4 x5 . ap (lam 6 (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) x1 (If_i (x7 = 2) x2 (If_i (x7 = 3) x3 (If_i (x7 = 4) x4 x5)))))) 2 = x2
Theorem Field_of_RealsStruct_2f^{Field_of_RealsStruct_2f} : ∀ x0 . RealsStruct x0 ⟶ (λ x2 . ap (ap (ap (Field_of_RealsStruct x0) 2) x2)) = field2b x0 (proof)
Theorem explicit_Field_of_RealsStruct_2^{explicit_Field_of_RealsStruct_2} : ∀ x0 . RealsStruct x0 ⟶ explicit_Field (field0 x0) (ap (Field_of_RealsStruct x0) 3) (ap (Field_of_RealsStruct x0) 4) (decode_b (ap (Field_of_RealsStruct x0) 1)) (decode_b (ap (Field_of_RealsStruct x0) 2)) (proof)
Definition Field^Field := λ x0 . and (struct_b_b_e_e x0) (unpack_b_b_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . explicit_Field x1 x4 x5 x2 x3))
Known prop_ext^prop_ext : ∀ x0 x1 : ο . iff x0 x1 ⟶ x0 = x1
Known iff_sym^iff_sym : ∀ x0 x1 : ο . iff x0 x1 ⟶ iff x1 x0
Known explicit_Field_repindep^{explicit_Field_repindep} : ∀ x0 x1 x2 . ∀ x3 x4 x5 x6 : ι → ι → ι . (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x7 x8 = x5 x7 x8) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x7 x8 = x6 x7 x8) ⟶ iff (explicit_Field x0 x1 x2 x3 x4) (explicit_Field x0 x1 x2 x5 x6)
Theorem Field_Field_of_RealsStruct^{Field_Field_of_RealsStruct} : ∀ x0 . RealsStruct x0 ⟶ Field (Field_of_RealsStruct x0) (proof)
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Theorem RealsStruct_minus_eq^{RealsStruct_minus_eq} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ Field_minus (Field_of_RealsStruct x0) x1 = explicit_Field_minus (field0 x0) (ap (Field_of_RealsStruct x0) 3) (ap (Field_of_RealsStruct x0) 4) (decode_b (ap (Field_of_RealsStruct x0) 1)) (decode_b (ap (Field_of_RealsStruct x0) 2)) x1 (proof)
Known explicit_Field_minus_clos^{explicit_Field_minus_clos} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x5 ∈ x0
Theorem RealsStruct_minus_clos^{RealsStruct_minus_clos} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ Field_minus (Field_of_RealsStruct x0) x1 ∈ field0 x0 (proof)
Known explicit_Field_minus_R^{explicit_Field_minus_R} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 x5 (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x1
Theorem RealsStruct_minus_R^{RealsStruct_minus_R} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field1b x0 x1 (Field_minus (Field_of_RealsStruct x0) x1) = field4 x0 (proof)
Theorem RealsStruct_minus_L^{RealsStruct_minus_L} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field1b x0 (Field_minus (Field_of_RealsStruct x0) x1) x1 = field4 x0 (proof)
Known explicit_Field_plus_cancelL^{explicit_Field_plus_cancelL} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 x6 = x3 x5 x7 ⟶ x6 = x7
Theorem RealsStruct_plus_cancelL^{RealsStruct_plus_cancelL} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field1b x0 x1 x2 = field1b x0 x1 x3 ⟶ x2 = x3 (proof)
Theorem RealsStruct_minus_eq2^{RealsStruct_minus_eq2} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ Field_minus (Field_of_RealsStruct x0) x1 = explicit_Field_minus (field0 x0) (field4 x0) (RealsStruct_one x0) (field1b x0) (field2b x0) x1 (proof)
Known explicit_Field_plus_cancelR^{explicit_Field_plus_cancelR} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 x7 = x3 x6 x7 ⟶ x5 = x6
Theorem RealsStruct_plus_cancelR^{RealsStruct_plus_cancelR} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field1b x0 x1 x3 = field1b x0 x2 x3 ⟶ x1 = x2 (proof)
Theorem RealsStruct_minus_invol^{RealsStruct_minus_invol} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ Field_minus (Field_of_RealsStruct x0) (Field_minus (Field_of_RealsStruct x0) x1) = x1 (proof)
Theorem RealsStruct_minus_one_In^{RealsStruct_minus_one_In} : ∀ x0 . RealsStruct x0 ⟶ Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0) ∈ field0 x0 (proof)
Known explicit_Field_zero_multR^{explicit_Field_zero_multR} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x5 x1 = x1
Theorem RealsStruct_zero_multR^{RealsStruct_zero_multR} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field2b x0 x1 (field4 x0) = field4 x0 (proof)
Theorem RealsStruct_zero_multL^{RealsStruct_zero_multL} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field2b x0 (field4 x0) x1 = field4 x0 (proof)
Theorem RealsStruct_minus_mult^{RealsStruct_minus_mult} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ Field_minus (Field_of_RealsStruct x0) x1 = field2b x0 (Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0)) x1 (proof)
Theorem RealsStruct_minus_one_square^{RealsStruct_minus_one_square} : ∀ x0 . RealsStruct x0 ⟶ field2b x0 (Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0)) (Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0)) = RealsStruct_one x0 (proof)
Known explicit_Field_minus_square^{explicit_Field_minus_square} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 (explicit_Field_minus x0 x1 x2 x3 x4 x5) (explicit_Field_minus x0 x1 x2 x3 x4 x5) = x4 x5 x5
Theorem RealsStruct_minus_square^{RealsStruct_minus_square} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field2b x0 (Field_minus (Field_of_RealsStruct x0) x1) (Field_minus (Field_of_RealsStruct x0) x1) = field2b x0 x1 x1 (proof)
Theorem RealsStruct_minus_zero^{RealsStruct_minus_zero} : ∀ x0 . RealsStruct x0 ⟶ Field_minus (Field_of_RealsStruct x0) (field4 x0) = field4 x0 (proof)
Known explicit_Field_dist_R^{explicit_Field_dist_R} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 (x3 x5 x6) x7 = x3 (x4 x5 x7) (x4 x6 x7)
Theorem RealsStruct_dist_R^{RealsStruct_dist_R} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field2b x0 (field1b x0 x1 x2) x3 = field1b x0 (field2b x0 x1 x3) (field2b x0 x2 x3) (proof)
Theorem RealsStruct_minus_plus_dist^{RealsStruct_minus_plus_dist} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ Field_minus (Field_of_RealsStruct x0) (field1b x0 x1 x2) = field1b x0 (Field_minus (Field_of_RealsStruct x0) x1) (Field_minus (Field_of_RealsStruct x0) x2) (proof)
Theorem RealsStruct_minus_mult_L^{RealsStruct_minus_mult_L} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field2b x0 (Field_minus (Field_of_RealsStruct x0) x1) x2 = Field_minus (Field_of_RealsStruct x0) (field2b x0 x1 x2) (proof)
Theorem RealsStruct_minus_mult_R^{RealsStruct_minus_mult_R} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field2b x0 x1 (Field_minus (Field_of_RealsStruct x0) x2) = Field_minus (Field_of_RealsStruct x0) (field2b x0 x1 x2) (proof)
Known explicit_Field_mult_zero_inv^{explicit_Field_mult_zero_inv} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = x1 ⟶ or (x5 = x1) (x6 = x1)
Theorem RealsStruct_mult_zero_inv^{RealsStruct_mult_zero_inv} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field2b x0 x1 x2 = field4 x0 ⟶ or (x1 = field4 x0) (x2 = field4 x0) (proof)
Theorem RealsStruct_square_zero_inv^{RealsStruct_square_zero_inv} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field2b x0 x1 x1 = field4 x0 ⟶ x1 = field4 x0 (proof)
Theorem RealsStruct_minus_leq^{RealsStruct_minus_leq} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_leq x0 x1 x2 ⟶ RealsStruct_leq x0 (Field_minus (Field_of_RealsStruct x0) x2) (Field_minus (Field_of_RealsStruct x0) x1) (proof)
Known explicit_OrderedField_square_nonneg^{explicit_OrderedField_square_nonneg} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x5 x1 (x4 x6 x6)
Theorem RealsStruct_square_nonneg^{RealsStruct_square_nonneg} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ RealsStruct_leq x0 (field4 x0) (field2b x0 x1 x1) (proof)
Known explicit_OrderedField_sum_squares_nonneg^{explicit_OrderedField_sum_squares_nonneg} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x5 x1 (x3 (x4 x6 x6) (x4 x7 x7))
Theorem RealsStruct_sum_squares_nonneg^{RealsStruct_sum_squares_nonneg} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_leq x0 (field4 x0) (field1b x0 (field2b x0 x1 x1) (field2b x0 x2 x2)) (proof)
Known explicit_OrderedField_sum_nonneg_zero_inv^{explicit_OrderedField_sum_nonneg_zero_inv} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x5 x1 x6 ⟶ x5 x1 x7 ⟶ x3 x6 x7 = x1 ⟶ and (x6 = x1) (x7 = x1)
Theorem RealsStruct_sum_nonneg_zero_inv^{RealsStruct_sum_nonneg_zero_inv} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_leq x0 (field4 x0) x1 ⟶ RealsStruct_leq x0 (field4 x0) x2 ⟶ field1b x0 x1 x2 = field4 x0 ⟶ and (x1 = field4 x0) (x2 = field4 x0) (proof)
Known explicit_OrderedField_sum_squares_zero_inv^{explicit_OrderedField_sum_squares_zero_inv} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 (x4 x6 x6) (x4 x7 x7) = x1 ⟶ and (x6 = x1) (x7 = x1)
Theorem RealsStruct_sum_squares_zero_inv^{RealsStruct_sum_squares_zero_inv} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field1b x0 (field2b x0 x1 x1) (field2b x0 x2 x2) = field4 x0 ⟶ and (x1 = field4 x0) (x2 = field4 x0) (proof)
Known explicit_OrderedField_leq_zero_one^{explicit_OrderedField_leq_zero_one} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ x5 x1 x2
Theorem RealsStruct_leq_zero_one^{RealsStruct_leq_zero_one} : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_leq x0 (field4 x0) (RealsStruct_one x0) (proof)
Param explicit_Nats^{explicit_Nats} : ι → ι → (ι → ι) → ο
Known explicit_Nats_natOfOrderedField^{explicit_Nats_natOfOrderedField} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Nats (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) x1 (λ x6 . x3 x6 x2)
Theorem RealsStruct_natOfOrderedField^{RealsStruct_natOfOrderedField} : ∀ x0 . RealsStruct x0 ⟶ explicit_Nats (RealsStruct_N x0) (field4 x0) (λ x1 . field1b x0 x1 (RealsStruct_one x0)) (proof)
Definition RealsStruct_Npos^{RealsStruct_Npos} := λ x0 . {x1 ∈ RealsStruct_N x0|x1 = field4 x0 ⟶ ∀ x2 : ο . x2}
Known explicit_PosNats_natOfOrderedField^{explicit_PosNats_natOfOrderedField} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Nats {x6 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x6 = x1 ⟶ ∀ x7 : ο . x7} x2 (λ x6 . x3 x6 x2)
Theorem RealsStruct_PosNats_natOfOrderedField^{RealsStruct_PosNats_natOfOrderedField} : ∀ x0 . RealsStruct x0 ⟶ explicit_Nats (RealsStruct_Npos x0) (RealsStruct_one x0) (λ x1 . field1b x0 x1 (RealsStruct_one x0)) (proof)
Definition RealsStruct_Z^{RealsStruct_Z} := λ x0 . {x1 ∈ field0 x0|or (or (Field_minus (Field_of_RealsStruct x0) x1 ∈ RealsStruct_Npos x0) (x1 = field4 x0)) (x1 ∈ RealsStruct_Npos x0)}
Definition 2a63f.. := λ x0 x1 . ∀ x2 : ο . (∀ x3 . and (x3 ∈ RealsStruct_Z x0) (∀ x4 : ο . (∀ x5 . and (x5 ∈ RealsStruct_Npos x0) (field2b x0 x5 x1 = x3) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2
Definition RealsStruct_Q^{RealsStruct_Q} := λ x0 . Sep (field0 x0) (2a63f.. x0)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param explicit_Nats_one_plus^{explicit_Nats_one_plus} : ι → ι → (ι → ι) → ι → ι → ι
Param explicit_Nats_one_mult^{explicit_Nats_one_mult} : ι → ι → (ι → ι) → ι → ι → ι
Known explicit_OrderedField_Npos_props^{explicit_OrderedField_Npos_props} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 : ο . ({x7 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x7 = x1 ⟶ ∀ x8 : ο . x8} ⊆ x0 ⟶ explicit_Nats {x7 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x7 = x1 ⟶ ∀ x8 : ο . x8} x2 (λ x7 . x3 x7 x2) ⟶ x2 ∈ {x7 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x7 = x1 ⟶ ∀ x8 : ο . x8} ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ x3 x7 x2 = x2 ⟶ ∀ x8 : ο . x8) ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ ∀ x8 : ι → ο . x8 x2 ⟶ (∀ x9 . x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11} ⟶ x8 (x3 x9 x2)) ⟶ x8 x7) ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ ∀ x8 . x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ explicit_Nats_one_plus {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11} x2 (λ x10 . x3 x10 x2) x7 x8 = x3 x7 x8) ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ ∀ x8 . x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ explicit_Nats_one_mult {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11} x2 (λ x10 . x3 x10 x2) x7 x8 = x4 x7 x8) ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ ∀ x8 . x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ x3 x7 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) ⟶ (∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ ∀ x8 . x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ x4 x7 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) ⟶ x6) ⟶ x6
Theorem RealsStruct_Npos_props^{RealsStruct_Npos_props} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 : ο . (RealsStruct_Npos x0 ⊆ field0 x0 ⟶ explicit_Nats (RealsStruct_Npos x0) (RealsStruct_one x0) (λ x2 . field1b x0 x2 (RealsStruct_one x0)) ⟶ RealsStruct_one x0 ∈ RealsStruct_Npos x0 ⟶ (∀ x2 . x2 ∈ RealsStruct_Npos x0 ⟶ field1b x0 x2 (RealsStruct_one x0) = RealsStruct_one x0 ⟶ ∀ x3 : ο . x3) ⟶ (∀ x2 . x2 ∈ RealsStruct_Npos x0 ⟶ ∀ x3 : ι → ο . x3 (RealsStruct_one x0) ⟶ (∀ x4 . x4 ∈ RealsStruct_Npos x0 ⟶ x3 (field1b x0 x4 (RealsStruct_one x0))) ⟶ x3 x2) ⟶ (∀ x2 . x2 ∈ RealsStruct_Npos x0 ⟶ ∀ x3 . x3 ∈ RealsStruct_Npos x0 ⟶ explicit_Nats_one_plus (RealsStruct_Npos x0) (RealsStruct_one x0) (λ x5 . field1b x0 x5 (RealsStruct_one x0)) x2 x3 = field1b x0 x2 x3) ⟶ (∀ x2 . x2 ∈ RealsStruct_Npos x0 ⟶ ∀ x3 . x3 ∈ RealsStruct_Npos x0 ⟶ explicit_Nats_one_mult (RealsStruct_Npos x0) (RealsStruct_one x0) (λ x5 . field1b x0 x5 (RealsStruct_one x0)) x2 x3 = field2b x0 x2 x3) ⟶ (∀ x2 . x2 ∈ RealsStruct_Npos x0 ⟶ ∀ x3 . x3 ∈ RealsStruct_Npos x0 ⟶ field1b x0 x2 x3 ∈ RealsStruct_Npos x0) ⟶ (∀ x2 . x2 ∈ RealsStruct_Npos x0 ⟶ ∀ x3 . x3 ∈ RealsStruct_Npos x0 ⟶ field2b x0 x2 x3 ∈ RealsStruct_Npos x0) ⟶ x1) ⟶ x1 (proof)
Theorem RealsStruct_Npos_R^{RealsStruct_Npos_R} : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_Npos x0 ⊆ field0 x0 (proof)
Theorem RealsStruct_one_Npos^{RealsStruct_one_Npos} : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_one x0 ∈ RealsStruct_Npos x0 (proof)
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem 35134.. : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_Z x0 = {x2 ∈ field0 x0|or (or (explicit_Field_minus (field0 x0) (field4 x0) (RealsStruct_one x0) (field1b x0) (field2b x0) x2 ∈ RealsStruct_Npos x0) (x2 = field4 x0)) (x2 ∈ RealsStruct_Npos x0)} (proof)
Known explicit_OrderedField_Z_props^{explicit_OrderedField_Z_props} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 : ο . ((∀ x7 . x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9} ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})}) ⟶ x1 ∈ {x7 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9}) (x7 = x1)) (x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9})} ⟶ {x7 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x7 = x1 ⟶ ∀ x8 : ο . x8} ⊆ {x7 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9}) (x7 = x1)) (x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9})} ⟶ {x7 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9}) (x7 = x1)) (x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9})} ⊆ x0 ⟶ (∀ x7 . x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})} ⟶ ∀ x8 : ο . (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ x8) ⟶ (x7 = x1 ⟶ x8) ⟶ (x7 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10} ⟶ x8) ⟶ x8) ⟶ x2 ∈ {x7 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9}) (x7 = x1)) (x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9})} ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x2 ∈ {x7 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9}) (x7 = x1)) (x7 ∈ {x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x8 = x1 ⟶ ∀ x9 : ο . x9})} ⟶ (∀ x7 . x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})} ⟶ explicit_Field_minus x0 x1 x2 x3 x4 x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})}) ⟶ (∀ x7 . x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})} ⟶ ∀ x8 . x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})} ⟶ x3 x7 x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})}) ⟶ (∀ x7 . x7 ∈ {x8 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10}) (x8 = x1)) (x8 ∈ {x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x9 = x1 ⟶ ∀ x10 : ο . x10})} ⟶ ∀ x8 . x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})} ⟶ x4 x7 x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})}) ⟶ x6) ⟶ x6
Theorem RealsStruct_Z_props^{RealsStruct_Z_props} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 : ο . ((∀ x2 . x2 ∈ RealsStruct_Npos x0 ⟶ Field_minus (Field_of_RealsStruct x0) x2 ∈ RealsStruct_Z x0) ⟶ field4 x0 ∈ RealsStruct_Z x0 ⟶ RealsStruct_Npos x0 ⊆ RealsStruct_Z x0 ⟶ RealsStruct_Z x0 ⊆ field0 x0 ⟶ (∀ x2 . x2 ∈ RealsStruct_Z x0 ⟶ ∀ x3 : ο . (Field_minus (Field_of_RealsStruct x0) x2 ∈ RealsStruct_Npos x0 ⟶ x3) ⟶ (x2 = field4 x0 ⟶ x3) ⟶ (x2 ∈ RealsStruct_Npos x0 ⟶ x3) ⟶ x3) ⟶ RealsStruct_one x0 ∈ RealsStruct_Z x0 ⟶ Field_minus (Field_of_RealsStruct x0) (RealsStruct_one x0) ∈ RealsStruct_Z x0 ⟶ (∀ x2 . x2 ∈ RealsStruct_Z x0 ⟶ Field_minus (Field_of_RealsStruct x0) x2 ∈ RealsStruct_Z x0) ⟶ (∀ x2 . x2 ∈ RealsStruct_Z x0 ⟶ ∀ x3 . x3 ∈ RealsStruct_Z x0 ⟶ field1b x0 x2 x3 ∈ RealsStruct_Z x0) ⟶ (∀ x2 . x2 ∈ RealsStruct_Z x0 ⟶ ∀ x3 . x3 ∈ RealsStruct_Z x0 ⟶ field2b x0 x2 x3 ∈ RealsStruct_Z x0) ⟶ x1) ⟶ x1 (proof)
Theorem 0eb6e.. : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_Q x0 = {x2 ∈ field0 x0|∀ x3 : ο . (∀ x4 . and (x4 ∈ {x5 ∈ field0 x0|or (or (explicit_Field_minus (field0 x0) (field4 x0) (RealsStruct_one x0) (field1b x0) (field2b x0) x5 ∈ RealsStruct_Npos x0) (x5 = field4 x0)) (x5 ∈ RealsStruct_Npos x0)}) (∀ x5 : ο . (∀ x6 . and (x6 ∈ RealsStruct_Npos x0) (field2b x0 x6 x2 = x4) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3} (proof)
Theorem RealsStruct_neg_Z^{RealsStruct_neg_Z} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ RealsStruct_Npos x0 ⟶ Field_minus (Field_of_RealsStruct x0) x1 ∈ RealsStruct_Z x0 (proof)
Theorem RealsStruct_zero_Z^{RealsStruct_zero_Z} : ∀ x0 . RealsStruct x0 ⟶ field4 x0 ∈ RealsStruct_Z x0 (proof)
Theorem RealsStruct_Npos_Z^{RealsStruct_Npos_Z} : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_Npos x0 ⊆ RealsStruct_Z x0 (proof)
Theorem RealsStruct_Z_R^{RealsStruct_Z_R} : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_Z x0 ⊆ field0 x0 (proof)
Definition explicit_OrderedField_rationalp^{explicit_OrderedField_rationalp} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . λ x6 . ∀ x7 : ο . (∀ x8 . and (x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})}) (∀ x9 : ο . (∀ x10 . and (x10 ∈ {x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x11 = x1 ⟶ ∀ x12 : ο . x12}) (x4 x10 x6 = x8) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7
Known explicit_OrderedField_Q_props^{explicit_OrderedField_Q_props} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 : ο . (Sep x0 (explicit_OrderedField_rationalp x0 x1 x2 x3 x4 x5) ⊆ x0 ⟶ (∀ x7 . x7 ∈ Sep x0 (explicit_OrderedField_rationalp x0 x1 x2 x3 x4 x5) ⟶ ∀ x8 : ο . (x7 ∈ x0 ⟶ ∀ x9 . x9 ∈ {x10 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x10 ∈ {x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x11 = x1 ⟶ ∀ x12 : ο . x12}) (x10 = x1)) (x10 ∈ {x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x11 = x1 ⟶ ∀ x12 : ο . x12})} ⟶ ∀ x10 . x10 ∈ {x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x11 = x1 ⟶ ∀ x12 : ο . x12} ⟶ x4 x10 x7 = x9 ⟶ x8) ⟶ x8) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ {x9 ∈ x0|or (or (explicit_Field_minus x0 x1 x2 x3 x4 x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11}) (x9 = x1)) (x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11})} ⟶ ∀ x9 . x9 ∈ {x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)|x10 = x1 ⟶ ∀ x11 : ο . x11} ⟶ x4 x9 x7 = x8 ⟶ x7 ∈ Sep x0 (explicit_OrderedField_rationalp x0 x1 x2 x3 x4 x5)) ⟶ x6) ⟶ x6
Theorem RealsStruct_Q_props^{RealsStruct_Q_props} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 : ο . (RealsStruct_Q x0 ⊆ field0 x0 ⟶ (∀ x2 . x2 ∈ RealsStruct_Q x0 ⟶ ∀ x3 : ο . (x2 ∈ field0 x0 ⟶ ∀ x4 . x4 ∈ RealsStruct_Z x0 ⟶ ∀ x5 . x5 ∈ RealsStruct_Npos x0 ⟶ field2b x0 x5 x2 = x4 ⟶ x3) ⟶ x3) ⟶ (∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ RealsStruct_Z x0 ⟶ ∀ x4 . x4 ∈ RealsStruct_Npos x0 ⟶ field2b x0 x4 x2 = x3 ⟶ x2 ∈ RealsStruct_Q x0) ⟶ x1) ⟶ x1 (proof)
Theorem RealsStruct_Q_R^{RealsStruct_Q_R} : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_Q x0 ⊆ field0 x0 (proof)
Theorem RealsStruct_Z_Q^{RealsStruct_Z_Q} : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_Z x0 ⊆ RealsStruct_Q x0 (proof)