let A, B be Ordinal; :: thesis: exp A,(succ B) = A *^ (exp A,B)
deffunc H₁( Ordinal) -> Ordinal = exp A,$1;
deffunc H₂( Ordinal, Ordinal-Sequence) -> Ordinal = lim $2;
deffunc H₃( Ordinal, Ordinal) -> Ordinal = A *^ $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 . {} = 1 & ( 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 Def20;
for B being Ordinal holds H₁( succ B) = H₃(B,H₁(B)) from ORDINAL2:sch 15(A1);
hence exp A,(succ B) = A *^ (exp A,B) ; :: thesis: verum