Let x0 of type ι be given.
Let x1 of type ι be given.
Apply setminusE with
omega,
1,
x0,
mul_nat x0 x1 ∈ setminus omega 1 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply setminusE with
omega,
1,
x1,
mul_nat x0 x1 ∈ setminus omega 1 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply setminusI with
omega,
1,
mul_nat x0 x1 leaving 2 subgoals.
Apply nat_p_omega with
mul_nat x0 x1.
Apply mul_nat_p with
x0,
x1 leaving 2 subgoals.
Apply omega_nat_p with
x0.
The subproof is completed by applying H2.
Apply omega_nat_p with
x1.
The subproof is completed by applying H4.
Apply cases_1 with
mul_nat x0 x1,
λ x2 . x2 = 0 leaving 2 subgoals.
The subproof is completed by applying H6.
Let x2 of type ι → ι → ο be given.
Assume H7: x2 0 0.
The subproof is completed by applying H7.
Apply unknownprop_2da221bcdd2314e7a8865e1e89957a529238abd39a22657b0cdfc26f16078944 with
x0,
x1,
False leaving 5 subgoals.
Apply omega_nat_p with
x0.
The subproof is completed by applying H2.
Apply omega_nat_p with
x1.
The subproof is completed by applying H4.
The subproof is completed by applying L7.
Assume H8: x0 = 0.
Apply H3.
Apply H8 with
λ x2 x3 . x3 ∈ 1.
The subproof is completed by applying In_0_1.
Assume H8: x1 = 0.
Apply H5.
Apply H8 with
λ x2 x3 . x3 ∈ 1.
The subproof is completed by applying In_0_1.