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Apply andI with 1 ∈ omega, ∀ x0 . x0 ∈ omega ⟶ 1 = mul_nat 2 x0 ⟶ ∀ x1 : ο . x1 leaving 2 subgoals.
Apply nat_p_omega with 1.
The subproof is completed by applying nat_1.
Let x0 of type ι be given.
Apply nat_inv with x0, False leaving 3 subgoals.
Apply omega_nat_p with x0.
The subproof is completed by applying H0.
Assume H2: x0 = 0.
Apply neq_1_0.
set y1 to be 1
set y2 to be 0
Claim L3: ∀ x3 : ι → ο . x3 y2 ⟶ x3 y1
Let x3 of type ι → ο be given.
Assume H3: x3 0.
Apply H1 with λ x4 . x3.
Apply H2 with λ x4 x5 . mul_nat 2 x5 = 0, λ x4 . x3 leaving 2 subgoals.
The subproof is completed by applying mul_nat_0R with 2.
The subproof is completed by applying H3.
Let x3 of type ι → ι → ο be given.
Apply L3 with λ x4 . x3 x4 y2 ⟶ x3 y2 x4.
Assume H4: x3 y2 y2.
The subproof is completed by applying H4.
Apply H2 with False.
Let x1 of type ι be given.
Apply H3 with False.
Apply In_irref with 1.
Apply H1 with λ x2 x3 . 1 ∈ x3.
Apply H5 with λ x2 x3 . 1 ∈ mul_nat 2 x3.
Apply mul_nat_SR with 2, x1, λ x2 x3 . 1 ∈ x3 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply mul_nat_p with 2, x1 leaving 2 subgoals.
The subproof is completed by applying nat_2.
The subproof is completed by applying H4.
Apply add_nat_SL with 1, mul_nat 2 x1, λ x2 x3 . 1 ∈ x3 leaving 3 subgoals.
The subproof is completed by applying nat_1.
The subproof is completed by applying L6.
Apply add_nat_SL with 0, mul_nat 2 x1, λ x2 x3 . 1 ∈ ordsucc x3 leaving 3 subgoals.
The subproof is completed by applying nat_0.
The subproof is completed by applying L6.
Apply add_nat_0L with mul_nat 2 x1, λ x2 x3 . 1 ∈ ordsucc (ordsucc x3) leaving 2 subgoals.
The subproof is completed by applying L6.
Apply nat_ordsucc_in_ordsucc with ordsucc (mul_nat 2 x1), 0 leaving 2 subgoals.
Apply nat_ordsucc with mul_nat 2 x1.
The subproof is completed by applying L6.
Apply nat_0_in_ordsucc with mul_nat 2 x1.
The subproof is completed by applying L6.
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