let A be Ordinal; :: thesis: A +^ {} = A
deffunc H₁( Ordinal) -> Ordinal = A +^ $1;
deffunc H₂( Ordinal, T-Sequence) -> Ordinal = sup $2;
deffunc H₃( Ordinal, Ordinal) -> set = succ $2;
A1: for B, C being Ordinal holds
( C = H₁(B) iff ex fi being Ordinal-Sequence st
( C = last fi & dom fi = succ B & fi . {} = A & ( for C being Ordinal st succ C in succ B holds
fi . (succ C) = H₃(C,fi . C) ) & ( for C being Ordinal st C in succ B & C <> {} & C is limit_ordinal holds
fi . C = H₂(C,fi | C) ) ) ) by Def18;
thus H₁( {} ) = A from ORDINAL2:sch 14(A1); :: thesis: verum