Let x0 of type ι be given.
Apply andI with
mul_nat 2 x0 ∈ omega,
∃ x1 . and (x1 ∈ omega) (mul_nat 2 x0 = mul_nat 2 x1) leaving 2 subgoals.
Apply nat_p_omega with
mul_nat 2 x0.
Apply mul_nat_p with
2,
x0 leaving 2 subgoals.
The subproof is completed by applying nat_2.
The subproof is completed by applying H0.
Let x1 of type ο be given.
Apply H1 with
x0.
Apply andI with
x0 ∈ omega,
mul_nat 2 x0 = mul_nat 2 x0 leaving 2 subgoals.
Apply nat_p_omega with
x0.
The subproof is completed by applying H0.
Let x2 of type ι → ι → ο be given.
The subproof is completed by applying H2.