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

theorem Th49:
  1 in B & A <> {} & A is limit_ordinal implies for fi st dom fi =
  A & for C st C in A holds fi.C = C*^B holds A*^B = sup fi
proof
  assume that
A1: 1 in B and
A2: A <> {} and
A3: A is limit_ordinal;
  let fi;
  assume that
A4: dom fi = A and
A5: for C st C in A holds fi.C = C*^B;
  now
    given C such that
A6: sup fi = succ C;
    consider D such that
A7: D in rng fi and
A8: C c= D by A6,ORDINAL1:6,ORDINAL2:21;
    D in sup fi by A7,ORDINAL2:19;
    then
A9: succ D c= succ C by A6,ORDINAL1:21;
    succ C c= succ D by A8,ORDINAL2:1;
    then succ C = succ D by A9;
    then C = D by ORDINAL1:7;
    then consider x being object such that
A10: x in dom fi and
A11: C = fi.x by A7,FUNCT_1:def 3;
    reconsider x as Ordinal by A10;
A12: C = x*^B by A4,A5,A10,A11;
    C+^1 in C+^B by A1,ORDINAL2:32;
    then
A13: sup fi in C+^B by A6,ORDINAL2:31;
A14: (succ x)*^B = x*^B+^B by ORDINAL2:36;
A15: succ x in dom fi by A3,A4,A10,ORDINAL1:28;
    then fi.succ x = (succ x)*^B by A4,A5;
    then C+^B in rng fi by A15,A12,A14,FUNCT_1:def 3;
    hence contradiction by A13,ORDINAL2:19;
  end;
  then
A16: sup fi is limit_ordinal by ORDINAL1:29;
  A*^B = union sup fi by A2,A3,A4,A5,ORDINAL2:37;
  hence thesis by A16;
end;
