let A be Ordinal; :: thesis: A +^ 0 = A
deffunc H₁( Ordinal, Ordinal) -> set = succ $2;
deffunc H₂( Ordinal, Sequence) -> Ordinal = sup $2;
deffunc H₃( Ordinal) -> Ordinal = A +^ $1;
A1: for B, C being Ordinal holds
( C = H₃(B) iff ex fi being Ordinal-Sequence st
( C = last fi & dom fi = succ B & fi . 0 = A & ( for C being Ordinal st succ C in succ B holds
fi . (succ C) = H₁(C,fi . C) ) & ( for C being Ordinal st C in succ B & C <> 0 & C is limit_ordinal holds
fi . C = H₂(C,fi | C) ) ) ) by Def14;
thus H₃( 0 ) = A from ORDINAL2:sch 14(A1); :: thesis: verum