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

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