reserve x,y,z for Element of REAL+;

theorem
  x <=' y implies x -' z <=' y -' z
proof
  assume
A1: x <=' y;
  per cases;
  suppose
A2: z <=' x;
    then z <=' y by A1,Th3;
    then
A3: y -' z + z = y by Def1;
    x -' z + z = x by A2,Def1;
    hence thesis by A1,A3,Th7;
  end;
  suppose
    not z <=' x;
    then x -' z = {} by Def1;
    hence thesis by Th6;
  end;
end;
