reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;

theorem Th41:
  A in B*^C & B is limit_ordinal implies ex D st D in B & A in D*^ C
proof
  assume that
A1: A in B*^C and
A2: B is limit_ordinal;
  deffunc F(Ordinal) = $1 *^ C;
  consider fi such that
A3: dom fi = B & for D st D in B holds fi.D = F(D) from ORDINAL2:sch 3;
  B <> {} by A1,ORDINAL2:35;
  then B*^C = union sup fi by A2,A3,ORDINAL2:37
    .= union sup rng fi;
  then consider X such that
A4: A in X and
A5: X in sup rng fi by A1,TARSKI:def 4;
  reconsider X as Ordinal by A5;
  consider D such that
A6: D in rng fi and
A7: X c= D by A5,ORDINAL2:21;
  consider x being object such that
A8: x in dom fi and
A9: D = fi.x by A6,FUNCT_1:def 3;
  reconsider x as Ordinal by A8;
  take E = succ x;
  thus E in B by A2,A3,A8,ORDINAL1:28;
A10: D+^{} = D by ORDINAL2:27;
A11: C <> {} by A1,ORDINAL2:38;
  E*^C = x*^C+^C by ORDINAL2:36
    .= D+^C by A3,A8,A9;
  then D in E*^C by A11,A10,Th8,ORDINAL2:32;
  hence thesis by A4,A7,ORDINAL1:10;
end;
