let A, B be Ordinal; ( 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 )
; 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; ( 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)
; 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); verum