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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: nat_p x0.
Assume H1: nat_p x1.
Assume H2: x0x1.
Apply nat_ind with λ x2 . add_nat x0 x2add_nat x1 x2 leaving 2 subgoals.
Apply add_nat_0R with x0, λ x2 x3 . x3add_nat x1 0.
Apply add_nat_0R with x1, λ x2 x3 . x0x3.
The subproof is completed by applying H2.
Let x2 of type ι be given.
Assume H3: nat_p x2.
Assume H4: add_nat x0 x2add_nat x1 x2.
Apply add_nat_SR with x0, x2, λ x3 x4 . x4add_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.
Claim L5: ordinal (add_nat x0 x2)
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.
Claim L6: ordinal (add_nat x1 x2)
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.