
theorem :: TimesM:
for x, y, z being Real st 0 <= x holds x*(y-'z) = x*y -' x*z
proof let x, y, z be Real;
 assume A1: x >= 0;
  per cases;
  suppose A2: y-z >= 0;
    A3: x*(y-z) >= 0 by A1,A2;
    thus x*(y-'z) = x*(y-z) by A2,XREAL_0:def 2
         .= x*y - x*z
         .= x*y -' x*z by A3,XREAL_0:def 2;
  end;
  suppose A4: y-z < 0;
    per cases by A1;
    suppose A5: x = 0;
      thus x*(y-'z) = x*y -' x*z by A5,XREAL_1:232;
    end;
    suppose A6: x > 0;
        x*(y-z) < 0 by A4,A6;
      then A7: x*y - x*z < 0;
     thus x*(y-'z) = x*0 by A4,XREAL_0:def 2
      .= x*y -' x*z by A7,XREAL_0:def 2;
    end;
  end;
end;
