
theorem LemmaIncreas:
  for f being bijective increasing UnOp of [.0,1.] holds
    f" is increasing
  proof
    set X = [.0,1.];
    let f be bijective increasing UnOp of [.0,1.];
A2: (f|X)" | (f.:X) is increasing by FC9;
A3: dom f = X & rng f = X by FUNCT_2:def 1,def 3;
    f.:X = X by RELAT_1:113,A3;
    hence thesis by A2;
  end;
