
theorem
  for f being decreasing bijective UnOp of [.0,1.] holds
    f.0 = 1 & f.1 = 0
  proof
    let f be decreasing bijective UnOp of [.0,1.];
KK: f.0 = 1
    proof
      set y = f.0;
      set X = [.0,1.];
K1:   1 in [.0,1.] by XXREAL_1:1;
      reconsider y as Element of [.0,1.] by XXREAL_1:1,FUNCT_2:5;
      assume
H0:   f.0 <> 1;
I3:   rng f = X by FUNCT_2:def 3;
      reconsider z = f".1 as Element of [.0,1.] by XXREAL_1:1,FUNCT_2:5;
L1:   f.z = 1 by FUNCT_1:35,I3,K1; then
      consider zz being Element of [.0,1.] such that
L2:   zz < z by Wazne1,H0;
      f.zz > f.z by L2,Decreas;
      hence contradiction by L1,XXREAL_1:1;
    end;
    f.1 = 0
    proof
      set X = [.0,1.];
K1:   0 in [.0,1.] by XXREAL_1:1;
      reconsider y = f.1 as Element of [.0,1.] by XXREAL_1:1,FUNCT_2:5;
      assume
H0:   f.1 <> 0;
I3:   rng f = X by FUNCT_2:def 3;
      reconsider z = f".0 as Element of [.0,1.] by XXREAL_1:1,FUNCT_2:5;
L1:   f.z = 0 by FUNCT_1:35,I3,K1; then
      consider zz being Element of [.0,1.] such that
L2:   zz > z by Wazne2,H0;
      f.zz < f.z by L2,Decreas;
      hence contradiction by L1,XXREAL_1:1;
    end;
    hence thesis by KK;
  end;
