let A be Ordinal; :: thesis: {} *^ A = {}
deffunc H₁( Ordinal) -> Ordinal = $1 *^ A;
deffunc H₂( Ordinal, T-Sequence) -> set = union (sup $2);
deffunc H₃( Ordinal, Ordinal) -> Ordinal = $2 +^ A;
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 . {} = {} & ( 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 Def19;
thus H₁( {} ) = {} from ORDINAL2:sch 14(A1); :: thesis: verum