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 = C *^ A ) holds
B *^ A = union (sup fi) )
deffunc H₁( Ordinal, Ordinal-Sequence) -> set = union (sup $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 = C *^ A ) holds
B *^ A = union (sup fi)
deffunc H₂( Ordinal, Ordinal) -> Ordinal = $2 +^ A;
deffunc H₃( Ordinal) -> Ordinal = $1 *^ A;
let fi be Ordinal-Sequence; :: thesis: ( dom fi = B & ( for C being Ordinal st C in B holds
fi . C = C *^ A ) implies B *^ A = union (sup fi) )
assume that
A2: dom fi = B and
A3: for C being Ordinal st C in B holds
fi . C = H₃(C) ; :: thesis: B *^ A = union (sup 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 = 0 & ( 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 Def15;
thus H₃(B) = H₁(B,fi) from ORDINAL2:sch 16(A4, A1, A2, A3); :: thesis: verum