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

theorem Th42:
  dom fi = dom psi & C <> {} & sup fi is limit_ordinal & (for A,B
  st A in dom fi & B = fi.A holds psi.A = B*^C) implies sup psi = (sup fi)*^C
proof
  assume that
A1: dom fi = dom psi and
A2: C <> {} and
A3: sup fi is limit_ordinal and
A4: for A,B st A in dom fi & B = fi.A holds psi.A = B*^C;
A5: (sup rng fi)*^C c= sup rng psi
  proof
    let x be object;
    assume
A6: x in (sup rng fi)*^C;
    then reconsider A = x as Ordinal;
    consider B such that
A7: B in sup rng fi and
A8: A in B*^C by A3,A6,Th41;
    consider D such that
A9: D in rng fi and
A10: B c= D by A7,ORDINAL2:21;
    consider y being object such that
A11: y in dom fi and
A12: D = fi.y by A9,FUNCT_1:def 3;
    reconsider y as Ordinal by A11;
    reconsider y9 = psi.y as Ordinal;
A13: y9 in rng psi by A1,A11,FUNCT_1:def 3;
    y9 = D*^C by A4,A11,A12;
    then
A14: D*^C in sup rng psi by A13,ORDINAL2:19;
    B*^C c= D*^C by A10,ORDINAL2:41;
    hence thesis by A8,A14,ORDINAL1:10;
  end;
  sup rng psi c= (sup rng fi)*^C
  proof
    let x be object;
    assume
A15: x in sup rng psi;
    then reconsider A = x as Ordinal;
    consider B such that
A16: B in rng psi and
A17: A c= B by A15,ORDINAL2:21;
    consider y being object such that
A18: y in dom psi and
A19: B = psi.y by A16,FUNCT_1:def 3;
    reconsider y as Ordinal by A18;
    reconsider y9 = fi.y as Ordinal;
    y9 in rng fi by A1,A18,FUNCT_1:def 3;
    then
A20: y9 in sup rng fi by ORDINAL2:19;
    B = y9*^C by A1,A4,A18,A19;
    then B in (sup rng fi)*^C by A2,A20,ORDINAL2:40;
    hence thesis by A17,ORDINAL1:12;
  end;
  hence thesis by A5;
end;
