Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H2: x0 ⊆ x1.
Apply nat_ind with
λ x2 . mul_nat x0 x2 ⊆ mul_nat x1 x2 leaving 2 subgoals.
Apply mul_nat_0R with
x0,
λ x2 x3 . x3 ⊆ mul_nat x1 0.
Apply mul_nat_0R with
x1,
λ x2 x3 . 0 ⊆ x3.
The subproof is completed by applying Subq_ref with 0.
Let x2 of type ι be given.
Apply mul_nat_SR with
x0,
x2,
λ x3 x4 . x4 ⊆ mul_nat x1 (ordsucc x2) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply mul_nat_SR with
x1,
x2,
λ x3 x4 . add_nat x0 (mul_nat x0 x2) ⊆ x4 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply Subq_tra with
add_nat x0 (mul_nat x0 x2),
add_nat x1 (mul_nat x0 x2),
add_nat x1 (mul_nat x1 x2) leaving 2 subgoals.
Apply unknownprop_d410a3a5a85dee8b88026a07b7ddf206621becbaaea4930e72baeb4af5fcafdc with
x0,
x1,
mul_nat x0 x2 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
Apply mul_nat_p with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Apply unknownprop_52447d35417df72c3930bddf586528d141c8a01b517a048173b612e2a2431a27 with
mul_nat x0 x2,
mul_nat x1 x2,
x1 leaving 4 subgoals.
Apply mul_nat_p with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Apply mul_nat_p with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H1.