let A, B be Ordinal; :: thesis: (succ B) *^ A = (B *^ A) +^ A
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;
for B being Ordinal holds H₂( succ B) = H₃(B,H₂(B)) from ORDINAL2:sch 15(A1);
hence (succ B) *^ A = (B *^ A) +^ A ; :: thesis: verum