let A be Ordinal; :: thesis: 0 *^ A = 0
deffunc H₁( Ordinal, Sequence) -> set = union (sup $2);
deffunc H₂( Ordinal) -> Ordinal = $1 *^ A;
deffunc H₃( Ordinal, Ordinal) -> Ordinal = $2 +^ A;
A1: for B, C being Ordinal holds
( C = H₂(B) iff ex fi being Ordinal-Sequence st
( C = last fi & dom fi = succ B & fi . 0 = 0 & ( 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 <> 0 & C is limit_ordinal holds
fi . C = H₁(C,fi | C) ) ) ) by Def15;
thus H₂( 0 ) = 0 from ORDINAL2:sch 14(A1); :: thesis: verum