
theorem NonInc:
  for f being UnOp of [.0,1.] holds
    f is non-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 non-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;
