Proofgold Address

Param ap^ap : ι → ι → ι
Definition field0^{RealsStruct_carrier} := λ x0 . ap x0 0
Definition decode_b^decode_b := λ x0 x1 . ap (ap x0 x1)
Param ordsucc^ordsucc : ι → ι
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
Param pack_b_b_e_e^pack_b_b_e_e : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι
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 If_i^If_i : ο → ι → ι → ι
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 pack_b_b_e_e_0_eq2^{pack_b_b_e_e_0_eq2} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x0 = ap (pack_b_b_e_e x0 x1 x2 x3 x4) 0
Theorem Field_of_RealsStruct_0^{Field_of_RealsStruct_0} : ∀ x0 . ap (Field_of_RealsStruct x0) 0 = field0 x0 (proof)
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
Theorem 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 (proof)
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
Theorem 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 (proof)
Known pack_b_b_e_e_3_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x3 = ap (pack_b_b_e_e x0 x1 x2 x3 x4) 3
Theorem Field_of_RealsStruct_3^{Field_of_RealsStruct_3} : ∀ x0 . ap (Field_of_RealsStruct x0) 3 = field4 x0 (proof)
Known pack_b_b_e_e_4_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x4 = ap (pack_b_b_e_e x0 x1 x2 x3 x4) 4
Theorem Field_of_RealsStruct_4^{Field_of_RealsStruct_4} : ∀ x0 . ap (Field_of_RealsStruct x0) 4 = RealsStruct_one x0 (proof)
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 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
Param explicit_Reals^{explicit_Reals} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Definition RealsStruct^RealsStruct := 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_Reals x1 x5 x6 x2 x3 x4))
Param pack_b_b_r_e_e^{pack_b_b_r_e_e} : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ι → ι
Known struct_b_b_r_e_e_eta^{struct_b_b_r_e_e_eta} : ∀ x0 . struct_b_b_r_e_e x0 ⟶ x0 = pack_b_b_r_e_e (ap x0 0) (decode_b (ap x0 1)) (decode_b (ap x0 2)) (decode_r (ap x0 3)) (ap x0 4) (ap x0 5)
Theorem 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) (proof)
Known RealsStruct_unpack_eq^{RealsStruct_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_Reals x7 x11 x12 x8 x9 x10) = explicit_Reals x0 x4 x5 x1 x2 x3
Theorem 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) (proof)
Known pack_struct_b_b_r_e_e_E4^{pack_struct_b_b_r_e_e_E4} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . struct_b_b_r_e_e (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) ⟶ x4 ∈ x0
Theorem RealsStruct_zero_In^{RealsStruct_zero_In} : ∀ x0 . RealsStruct x0 ⟶ field4 x0 ∈ field0 x0 (proof)
Known pack_struct_b_b_r_e_e_E5^{pack_struct_b_b_r_e_e_E5} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . struct_b_b_r_e_e (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) ⟶ x5 ∈ x0
Theorem RealsStruct_one_In^{RealsStruct_one_In} : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_one x0 ∈ field0 x0 (proof)
Known pack_struct_b_b_r_e_e_E1^{pack_struct_b_b_r_e_e_E1} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . struct_b_b_r_e_e (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x1 x6 x7 ∈ x0
Theorem RealsStruct_plus_clos^{RealsStruct_plus_clos} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field1b x0 x1 x2 ∈ field0 x0 (proof)
Known pack_struct_b_b_r_e_e_E2^{pack_struct_b_b_r_e_e_E2} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . struct_b_b_r_e_e (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x6 x7 ∈ x0
Theorem RealsStruct_mult_clos^{RealsStruct_mult_clos} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field2b x0 x1 x2 ∈ field0 x0 (proof)
Param explicit_OrderedField^{explicit_OrderedField} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Definition lt^lt := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . λ x6 x7 . and (x5 x6 x7) (x6 = x7 ⟶ ∀ x8 : ο . x8)
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 explicit_Field^{explicit_Field} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Param iff^iff : ο → ο → ο
Param or^or : 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
Known explicit_Field_E^{explicit_Field_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x6 (x3 x7 x8) = x3 (x3 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 = x3 x7 x6) ⟶ x1 ∈ x0 ⟶ (∀ x6 . x6 ∈ x0 ⟶ x3 x1 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (x3 x6 x8 = x1) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x4 x7 x8) = x4 (x4 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 = x4 x7 x6) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x6 : ο . x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ x4 x2 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ (x6 = x1 ⟶ ∀ x7 : ο . x7) ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (x4 x6 x8 = x2) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x3 x7 x8) = x3 (x4 x6 x7) (x4 x6 x8)) ⟶ x5) ⟶ explicit_Field x0 x1 x2 x3 x4 ⟶ x5
Theorem 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 (proof)
Theorem 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 (proof)
Theorem RealsStruct_zero_L^{RealsStruct_zero_L} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field1b x0 (field4 x0) x1 = x1 (proof)
Theorem 1c11b.. : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ field0 x0) (field1b x0 x1 x3 = field4 x0) ⟶ x2) ⟶ x2 (proof)
Theorem 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 (proof)
Theorem 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 (proof)
Theorem RealsStruct_one_neq_zero^{RealsStruct_one_neq_zero} : ∀ x0 . RealsStruct x0 ⟶ RealsStruct_one x0 = field4 x0 ⟶ ∀ x1 : ο . x1 (proof)
Theorem RealsStruct_one_L^{RealsStruct_one_L} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field2b x0 (RealsStruct_one x0) x1 = x1 (proof)
Theorem c2c76.. : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ (x1 = field4 x0 ⟶ ∀ x2 : ο . x2) ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ field0 x0) (field2b x0 x1 x3 = RealsStruct_one x0) ⟶ x2) ⟶ x2 (proof)
Theorem 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) (proof)
Known explicit_OrderedField_leq_refl^{explicit_OrderedField_leq_refl} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x5 x6 x6
Theorem RealsStruct_leq_refl^{RealsStruct_leq_refl} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ RealsStruct_leq x0 x1 x1 (proof)
Known explicit_OrderedField_leq_antisym^{explicit_OrderedField_leq_antisym} : ∀ 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 x6 x7 ⟶ x5 x7 x6 ⟶ x6 = x7
Theorem RealsStruct_leq_antisym^{RealsStruct_leq_antisym} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_leq x0 x1 x2 ⟶ RealsStruct_leq x0 x2 x1 ⟶ x1 = x2 (proof)
Theorem RealsStruct_leq_tra^{RealsStruct_leq_tra} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ RealsStruct_leq x0 x1 x2 ⟶ RealsStruct_leq x0 x2 x3 ⟶ RealsStruct_leq x0 x1 x3 (proof)
Theorem 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) (proof)
Theorem 97dc1.. : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_leq x0 (field4 x0) x1 ⟶ RealsStruct_leq x0 (field4 x0) x2 ⟶ RealsStruct_leq x0 (field4 x0) (field2b x0 x1 x2) (proof)
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))
Theorem 76b2d.. : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ e08de.. x0 x0 (field4 x0) x1 ⟶ RealsStruct_leq x0 (field4 x0) x2 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ RealsStruct_N x0) (RealsStruct_leq x0 x2 (field2b x0 x4 x1)) ⟶ x3) ⟶ x3 (proof)
Theorem RealsStruct_Compl^{RealsStruct_Compl} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ setexp (field0 x0) (RealsStruct_N x0) ⟶ ∀ x2 . x2 ∈ setexp (field0 x0) (RealsStruct_N x0) ⟶ (∀ x3 . x3 ∈ RealsStruct_N x0 ⟶ and (and (RealsStruct_leq x0 (ap x1 x3) (ap x2 x3)) (RealsStruct_leq x0 (ap x1 x3) (ap x1 (field1b x0 x3 (RealsStruct_one x0))))) (RealsStruct_leq x0 (ap x2 (field1b x0 x3 (RealsStruct_one x0))) (ap x2 x3))) ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ field0 x0) (∀ x5 . x5 ∈ RealsStruct_N x0 ⟶ and (RealsStruct_leq x0 (ap x1 x5) x4) (RealsStruct_leq x0 x4 (ap x2 x5))) ⟶ x3) ⟶ x3 (proof)
Theorem explicit_Field_of_RealsStruct^{explicit_Field_of_RealsStruct} : ∀ x0 . RealsStruct x0 ⟶ explicit_Field (field0 x0) (field4 x0) (RealsStruct_one x0) (field1b x0) (field2b x0) (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 andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
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)