Apply nat_inv_impred with
λ x0 . ∀ x1 . nat_p x1 ⟶ 0 ∈ x0 ⟶ 1 ∈ x1 ⟶ x0 ∈ mul_nat x0 x1 leaving 2 subgoals.
Let x0 of type ι be given.
Assume H1: 0 ∈ 0.
Apply FalseE with
1 ∈ x0 ⟶ 0 ∈ mul_nat 0 x0.
Apply EmptyE with
0.
The subproof is completed by applying H1.
Let x0 of type ι be given.
Apply nat_inv_impred with
λ x1 . 0 ∈ ordsucc x0 ⟶ 1 ∈ x1 ⟶ ordsucc x0 ∈ mul_nat (ordsucc x0) x1 leaving 2 subgoals.
Assume H2: 1 ∈ 0.
Apply FalseE with
ordsucc x0 ∈ mul_nat (ordsucc x0) 0.
Apply In_no2cycle with
0,
1 leaving 2 subgoals.
The subproof is completed by applying In_0_1.
The subproof is completed by applying H2.
Apply nat_inv_impred with
λ x1 . 0 ∈ ordsucc x0 ⟶ 1 ∈ ordsucc x1 ⟶ ordsucc x0 ∈ mul_nat (ordsucc x0) (ordsucc x1) leaving 2 subgoals.
Assume H2: 1 ∈ 1.
Apply FalseE with
ordsucc x0 ∈ mul_nat (ordsucc x0) 1.
Apply In_irref with
1.
The subproof is completed by applying H2.
Let x1 of type ι be given.
Apply mul_nat_SR with
ordsucc x0,
ordsucc x1,
λ x2 x3 . ordsucc x0 ∈ x3 leaving 2 subgoals.
Apply nat_ordsucc with
x1.
The subproof is completed by applying H1.
Apply add_nat_0R with
ordsucc x0,
λ x2 x3 . x2 ∈ add_nat (ordsucc x0) (mul_nat (ordsucc x0) (ordsucc x1)).
Apply unknownprop_0572fd07b7b27d4a8020f07cf9f53538e7a9507efdd502126aa0026898fbc5e2 with
ordsucc x0,
mul_nat (ordsucc x0) (ordsucc x1),
0 leaving 3 subgoals.
Apply nat_ordsucc with
x0.
The subproof is completed by applying H0.
Apply mul_nat_p with
ordsucc x0,
ordsucc x1 leaving 2 subgoals.
Apply nat_ordsucc with
x0.
The subproof is completed by applying H0.
Apply nat_ordsucc with
x1.
The subproof is completed by applying H1.
Apply mul_nat_SL with
x0,
ordsucc x1,
λ x2 x3 . 0 ∈ x3 leaving 3 subgoals.
The subproof is completed by applying H0.
Apply nat_ordsucc with
x1.
The subproof is completed by applying H1.
Apply add_nat_SR with
mul_nat x0 (ordsucc x1),
x1,
λ x2 x3 . 0 ∈ x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply nat_0_in_ordsucc with
add_nat (mul_nat x0 (ordsucc x1)) x1.
Apply add_nat_p with
mul_nat x0 (ordsucc x1),
x1 leaving 2 subgoals.
Apply mul_nat_p with
x0,
ordsucc x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply nat_ordsucc with
x1.
The subproof is completed by applying H1.
The subproof is completed by applying H1.