
theorem RosnieI:
  for f being UnOp of [.0,1.] holds
    f is increasing iff
      for a,b being Element of [.0,1.] st a < b holds f.a < f.b
  proof
    let f be UnOp of [.0,1.];
    dom f = [.0,1.] by FUNCT_2:def 1;
    hence f is increasing implies
      for a,b being Element of [.0,1.] st a < b holds f.a < f.b;
    assume
B1: for a,b being Element of [.0,1.] st a < b holds f.a < f.b;
    let e1,e2 be ExtReal;
    assume
B2: e1 in dom f & e2 in dom f & e1 < e2; then
    reconsider ee1 = e1, ee2 = e2 as Element of [.0,1.] by FUNCT_2:def 1;
    ee1 < ee2 by B2;
    hence thesis by B1;
  end;
