reserve A,A1,A2,B,C,D for Ordinal,
  X,Y for set,
  x,y,a,b,c for object,
  L,L1,L2,L3 for Sequence,
  f for Function;
reserve fi,psi for Ordinal-Sequence;

theorem
  A c= B implies A*^C c= B*^C
proof
  assume
A1: A c= B;
A2: now
    assume
A3: C <> 0;
    now
      assume A <> B;
      then A c< B by A1;
      then A*^C in B*^C by A3,Th40,ORDINAL1:11;
      hence thesis by ORDINAL1:def 2;
    end;
    hence thesis;
  end;
  C = 0 implies thesis by Th38;
  hence thesis by A2;
end;
