
theorem
  for f being continuous one-to-one Function of R^1, R^1 holds (for x, y
being Point of I[01], p, q, fx, fy being Real
   st x = p & y = q & p < q & fx = f
  .x & fy = f.y holds fx < fy) or
  for x, y being Point of I[01], p, q, fx, fy being Real
    st x = p & y = q & p < q & fx = f.x & fy = f.y holds fx > fy
proof
  let f be continuous one-to-one Function of R^1, R^1;
A1: [.0,1.] /\ dom f = [.0,1.] /\ the carrier of R^1 by FUNCT_2:def 1
    .= [.0,1.] by BORSUK_1:1,40,TOPMETR:20,XBOOLE_1:28;
  reconsider g = f as PartFunc of REAL, REAL by TOPMETR:17;
  per cases by Lm5;
  suppose
    g| [.0,1.] is increasing;
    hence thesis by A1,BORSUK_1:40,RFUNCT_2:20;
  end;
  suppose
    g| [.0,1.] is decreasing;
    hence thesis by A1,BORSUK_1:40,RFUNCT_2:21;
  end;
end;
