reserve n for Element of NAT,
  a, b for Real;

theorem
  a <= b implies [.a,b.] =
  { (1-l)*a + l*b where l is Real: 0 <= l & l <= 1 }
proof
  set X = {(1-l)*a+l*b where l is Real: 0 <= l & l <= 1};
  assume
A1: a <= b;
  hereby
    let x be object;
    assume
A2: x in [.a,b.];
    then reconsider y = x as Real;
A3: a <= y & y <= b by A2,XXREAL_1:1;
    per cases by A1,XXREAL_0:1;
    suppose
      a < b;
      then
A4:   b-a > b-b by XREAL_1:15;
      reconsider l = (y-a)/(b-a) as Real;
      l in the carrier of Closed-Interval-TSpace (0,1) by A3,BORSUK_6:2;
      then l in [.0,1.] by TOPMETR:18;
      then
A5:   0 <= l & l <= 1 by XXREAL_1:1;
      (1-l)*a+l*b = a+l*(b-a) .= a+(y-a) by A4,XCMPLX_1:87
        .= y;
      hence x in X by A5;
    end;
    suppose
      a = b;
      then (1-1)*a+1*b = y by A3,XXREAL_0:1;
      hence x in X;
    end;
  end;
  let x be object;
  assume x in X;
  then consider l being Real such that
A6: x = (1-l)*a+l*b and
A7: 0 <= l & l <= 1;
  a <= (1-l)*a+l*b & (1-l)*a+l*b <= b by A1,A7,XREAL_1:172,173;
  hence thesis by A6,XXREAL_1:1;
end;
