let A, B be Ordinal; :: thesis: ( B <> 0 & B is limit_ordinal implies for fi being Ordinal-Sequence st dom fi = B & ( for C being Ordinal st C in B holds
fi . C = exp (A,C) ) holds
exp (A,B) = lim fi )
deffunc H₁( Ordinal, Ordinal-Sequence) -> Ordinal = lim $2;
assume A1: ( B <> 0 & B is limit_ordinal ) ; :: thesis: for fi being Ordinal-Sequence st dom fi = B & ( for C being Ordinal st C in B holds
fi . C = exp (A,C) ) holds
exp (A,B) = lim fi
deffunc H₂( Ordinal, Ordinal) -> Ordinal = A *^ $2;
deffunc H₃( Ordinal) -> Ordinal = exp (A,$1);
let fi be Ordinal-Sequence; :: thesis: ( dom fi = B & ( for C being Ordinal st C in B holds
fi . C = exp (A,C) ) implies exp (A,B) = lim fi )
assume that
A2: dom fi = B and
A3: for C being Ordinal st C in B holds
fi . C = H₃(C) ; :: thesis: exp (A,B) = lim fi
A4: for B, C being Ordinal holds
( C = H₃(B) iff ex fi being Ordinal-Sequence st
( C = last fi & dom fi = succ B & fi . 0 = 1 & ( for C being Ordinal st succ C in succ B holds
fi . (succ C) = H₂(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 Def16;
thus H₃(B) = H₁(B,fi) from ORDINAL2:sch 16(A4, A1, A2, A3); :: thesis: verum