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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.
Claim L2: ordinal x0
Apply nat_p_ordinal with x0.
The subproof is completed by applying H0.
Claim L3: SNo x0
Apply nat_p_SNo with x0.
The subproof is completed by applying H0.
Claim L4: ordinal x1
Apply nat_p_ordinal with x1.
The subproof is completed by applying H1.
Assume H5: SNoLt x0 x1.
Apply add_SNo_com with x0, 1, λ x2 x3 . SNoLe x3 x1 leaving 3 subgoals.
The subproof is completed by applying L3.
The subproof is completed by applying SNo_1.
Apply ordinal_ordsucc_SNo_eq with x0, λ x2 x3 . SNoLe x2 x1 leaving 2 subgoals.
The subproof is completed by applying L2.
Apply ordinal_SNoLev_max_2 with x1, ordsucc x0 leaving 3 subgoals.
The subproof is completed by applying L4.
Apply nat_p_SNo with ordsucc x0.
Apply nat_ordsucc with x0.
The subproof is completed by applying H0.
Apply ordinal_SNoLev with ordsucc x0, λ x2 x3 . x3ordsucc x1 leaving 2 subgoals.
Apply ordinal_ordsucc with x0.
The subproof is completed by applying L2.
Apply ordinal_ordsucc_In with x1, x0 leaving 2 subgoals.
The subproof is completed by applying L4.
Apply ordinal_SNoLt_In with x0, x1 leaving 3 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying L4.
The subproof is completed by applying H5.