let B, A be Ordinal; :: thesis: ( B <> {} & 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 )
assume A1: ( B <> {} & 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 H₁( Ordinal) -> Ordinal = exp A,$1;
deffunc H₂( Ordinal, Ordinal-Sequence) -> Ordinal = lim $2;
deffunc H₃( Ordinal, Ordinal) -> Ordinal = A *^ $2;
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 = H₁(C) ; :: thesis: exp A,B = lim fi
A4: 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;
thus H₁(B) = H₂(B,fi) from ORDINAL2:sch 16(A4, A1, A2, A3); :: thesis: verum