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

theorem Th48:
  for C1,D1,C2,D2 being Ordinal st C1*^A+^D1 = C2*^A+^D2 & D1 in A
  & D2 in A holds C1 = C2 & D1 = D2
proof
  let C1,D1,C2,D2 be Ordinal such that
A1: C1*^A+^D1 = C2*^A+^D2 and
A2: D1 in A and
A3: D2 in A;
  set B = C1*^A+^D1;
A4: now
    assume C2 in C1;
    then consider C such that
A5: C1 = C2+^C and
A6: C <> {} by Th28;
    B = C2*^A+^C*^A+^D1 by A5,Th46
      .= C2*^A+^(C*^A+^D1) by Th30;
    then
A7: D2 = C*^A+^D1 by A1,Th21;
A8: C*^A c= C*^A+^D1 by Th24;
    A c= C*^A by A6,Th36;
    hence contradiction by A3,A7,A8,ORDINAL1:5;
  end;
  now
    assume C1 in C2;
    then consider C such that
A9: C2 = C1+^C and
A10: C <> {} by Th28;
    B = C1*^A+^C*^A+^D2 by A1,A9,Th46
      .= C1*^A+^(C*^A+^D2) by Th30;
    then
A11: D1 = C*^A+^D2 by Th21;
A12: C*^A c= C*^A+^D2 by Th24;
    A c= C*^A by A10,Th36;
    hence contradiction by A2,A11,A12,ORDINAL1:5;
  end;
  hence C1 = C2 by A4,ORDINAL1:14;
  hence thesis by A1,Th21;
end;
