Let x0 of type ι be given.
Let x1 of type ι be given.
Apply mul_nat_mul_SNo with
2,
x0,
λ x2 x3 . x3 = mul_nat 2 x1 ⟶ x0 = x1 leaving 3 subgoals.
Apply nat_p_omega with
2.
The subproof is completed by applying nat_2.
Apply nat_p_omega with
x0.
The subproof is completed by applying H0.
Apply mul_nat_mul_SNo with
2,
x1,
λ x2 x3 . mul_SNo 2 x0 = x3 ⟶ x0 = x1 leaving 3 subgoals.
Apply nat_p_omega with
2.
The subproof is completed by applying nat_2.
Apply nat_p_omega with
x1.
The subproof is completed by applying H1.
Apply mul_SNo_oneL with
x0,
λ x2 x3 . x2 = x1 leaving 2 subgoals.
Apply nat_p_SNo with
x0.
The subproof is completed by applying H0.
Apply mul_SNo_oneL with
x1,
λ x2 x3 . mul_SNo 1 x0 = x2 leaving 2 subgoals.
Apply nat_p_SNo with
x1.
The subproof is completed by applying H1.
Apply eps_1_half_eq3 with
λ x2 x3 . mul_SNo x2 x0 = mul_SNo x2 x1.
Apply mul_SNo_assoc with
eps_ 1,
2,
x0,
λ x2 x3 . x2 = mul_SNo (mul_SNo (eps_ 1) 2) x1 leaving 4 subgoals.
Apply SNo_eps_ with
1.
Apply nat_p_omega with
1.
The subproof is completed by applying nat_1.
Apply nat_p_SNo with
2.
The subproof is completed by applying nat_2.
Apply nat_p_SNo with
x0.
The subproof is completed by applying H0.
Apply mul_SNo_assoc with
eps_ 1,
2,
x1,
λ x2 x3 . mul_SNo (eps_ 1) (mul_SNo 2 x0) = x2 leaving 4 subgoals.
Apply SNo_eps_ with
1.
Apply nat_p_omega with
1.
The subproof is completed by applying nat_1.
Apply nat_p_SNo with
2.
The subproof is completed by applying nat_2.
Apply nat_p_SNo with
x1.
The subproof is completed by applying H1.
Apply H2 with
λ x2 x3 . mul_SNo (eps_ 1) x3 = mul_SNo (eps_ 1) (mul_SNo 2 x1).
Let x2 of type ι → ι → ο be given.
The subproof is completed by applying H3.