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

theorem Th7:
  x <=' y iff x + z <=' y + z
proof
  thus x <=' y implies x + z <=' y + z
  proof
    assume x <=' y;
    then consider z0 being Element of REAL+ such that
A1: x + z0 = y by ARYTM_2:9;
    x + z + z0 = y + z by A1,ARYTM_2:6;
    hence thesis by ARYTM_2:19;
  end;
  assume x + z <=' y + z;
  then consider z0 being Element of REAL+ such that
A2: x + z + z0 = y + z by ARYTM_2:9;
  y + z = x + z0 + z by A2,ARYTM_2:6;
  hence thesis by ARYTM_2:11,19;
end;
