theorem Th15:
  for xi being Ordinal-Sequence holds (C+^xi)|A = C+^(xi|A)
proof
  let xi be Ordinal-Sequence;
A1: dom (xi|A) = dom xi /\ A by RELAT_1:61;
A2: dom (C+^xi) = dom xi by ORDINAL3:def 1;
A3: dom ((C+^xi)|A) = dom (C+^xi) /\ A by RELAT_1:61;
A4: now
    let e be object;
    assume
A5: e in dom ((C+^xi)|A);
    then reconsider a = e as Ordinal;
A6: e in dom xi by A3,A2,A5,XBOOLE_0:def 4;
    thus ((C+^xi)|A).e = (C+^xi).a by A5,FUNCT_1:47
      .= C+^(xi.a) by A6,ORDINAL3:def 1
      .= C+^((xi|A).a) by A3,A1,A2,A5,FUNCT_1:47
      .= (C+^(xi|A)).e by A3,A1,A2,A5,ORDINAL3:def 1;
  end;
  dom (C+^(xi|A)) = dom (xi|A) by ORDINAL3:def 1;
  hence thesis by A1,A2,A4,FUNCT_1:2,RELAT_1:61;
end;
