Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H2: x0 ⊆ x1.
Apply nat_ind with
λ x2 . add_nat x0 x2 ⊆ add_nat x1 x2 leaving 2 subgoals.
Apply add_nat_0R with
x0,
λ x2 x3 . x3 ⊆ add_nat x1 0.
Apply add_nat_0R with
x1,
λ x2 x3 . x0 ⊆ x3.
The subproof is completed by applying H2.
Let x2 of type ι be given.
Apply add_nat_SR with
x0,
x2,
λ x3 x4 . x4 ⊆ add_nat x1 (ordsucc x2) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply add_nat_SR with
x1,
x2,
λ x3 x4 . ordsucc (add_nat x0 x2) ⊆ x4 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply nat_p_ordinal with
add_nat x0 x2.
Apply add_nat_p with
x0,
x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
Apply nat_p_ordinal with
add_nat x1 x2.
Apply add_nat_p with
x1,
x2 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
Apply unknownprop_7efa9b0eb4a7672f89f79f79d5dfd89fdd189d47f4be668ddc1bdc4223ecb821 with
add_nat x0 x2,
add_nat x1 x2 leaving 3 subgoals.
The subproof is completed by applying L5.
The subproof is completed by applying L6.
The subproof is completed by applying H4.