
theorem Th2:
  for a, b, x being Real st a <= x & x <= b holds (x - a) /
  (b - a) in the carrier of Closed-Interval-TSpace (0,1)
proof
  let a, b, x be Real;
  assume that
A1: a <= x and
A2: x <= b;
A3: a <= b by A1,A2,XXREAL_0:2;
A4: x - a <= b - a by A2,XREAL_1:9;
A5: (x - a) / (b - a) <= 1
  proof
    per cases by A3,XREAL_1:48;
    suppose
      b - a = 0;
      hence thesis by XCMPLX_1:49;
    end;
    suppose
      b - a > 0;
      hence thesis by A4,XREAL_1:185;
    end;
  end;
A6: x - a >= 0 by A1,XREAL_1:48;
  b - a >= 0 by A3,XREAL_1:48;
  then (x - a) / (b - a) in [.0,1.] by A5,A6,XXREAL_1:1;
  hence thesis by TOPMETR:18;
end;
