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

theorem Th39:
  {} <> dom fi & dom fi = dom psi & (for A,B st A in dom fi & B =
  fi.A holds psi.A = C+^B) implies sup psi = C+^sup fi
proof
  assume that
A1: {} <> dom fi and
A2: dom fi = dom psi and
A3: for A,B st A in dom fi & B = fi.A holds psi.A = C+^B;
  set z = the Element of dom fi;
  reconsider z9 = fi.z as Ordinal;
A4: C+^sup rng fi c= sup rng psi
  proof
    let x be object;
    assume
A5: x in C+^sup rng fi;
    then reconsider A = x as Ordinal;
A6: now
      given B such that
A7:   B in sup rng fi and
A8:   A = C+^B;
      consider D such that
A9:   D in rng fi and
A10:  B c= D by A7,ORDINAL2:21;
      consider x being object such that
A11:  x in dom fi and
A12:  D = fi.x by A9,FUNCT_1:def 3;
      reconsider x as Ordinal by A11;
      psi.x = C+^D by A3,A11,A12;
      then C+^D in rng psi by A2,A11,FUNCT_1:def 3;
      then C+^D in sup rng psi by ORDINAL2:19;
      hence A in sup rng psi by A8,A10,ORDINAL1:12,ORDINAL2:33;
    end;
    now
      C+^z9 = psi.z by A1,A3;
      then C+^z9 in rng psi by A1,A2,FUNCT_1:def 3;
      then
A13:  C+^z9 in sup rng psi by ORDINAL2:19;
      assume
A14:  A in C;
      C c= C+^z9 by Th24;
      then A c= C+^z9 by A14,ORDINAL1:def 2;
      hence A in sup rng psi by A13,ORDINAL1:12;
    end;
    hence thesis by A5,A6,Th38;
  end;
  sup rng psi c= C+^sup rng fi
  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 A2,A18,FUNCT_1:def 3;
    then
A20: y9 in sup rng fi by ORDINAL2:19;
    B = C+^y9 by A2,A3,A18,A19;
    then B in C+^sup rng fi by A20,ORDINAL2:32;
    hence thesis by A17,ORDINAL1:12;
  end;
  hence thesis by A4;
end;
