
theorem Th7:
  for x,y,z being Element of REAL+ st x <> {} & x *' y = x *' z holds y = z
proof
  let x,y,z be Element of REAL+;
  assume that
A1: x <> {} and
A2: x *' y = x *' z;
  per cases;
  suppose
A3: z <=' y;
    then x *' (y -' z) = (x *' y) - (x *' z) by ARYTM_1:26
      .= {} by A2,ARYTM_1:18;
    then {} = y -' z by A1,ARYTM_1:2
      .= y - z by A3,ARYTM_1:def 2;
    hence thesis by Th6;
  end;
  suppose
A4: y <=' z;
    then x *' (z -' y) = x *' z - x *' y by ARYTM_1:26
      .= {} by A2,ARYTM_1:18;
    then {} = z -' y by A1,ARYTM_1:2
      .= z - y by A4,ARYTM_1:def 2;
    hence thesis by Th6;
  end;
end;
