
theorem
  FNegation I_FD = N_CC
  proof
    set I = I_FD;
    set f = FNegation I;
    set g = N_CC;
A1: 0 in [.0,1.] by XXREAL_1:1;
    for x being Element of [.0,1.] holds f.x = g.x
    proof
      let x be Element of [.0,1.];
      1 - x in [.0,1.] by FUZNORM1:7; then
A2:   1 - x >= 0 by XXREAL_1:1;
      per cases;
      suppose
        x <= 0; then
B2:     x = 0 by XXREAL_1:1;
        f.x = I.(x,0) by FNeg
           .= 1 - x by FUZIMPL1:def 23,B2
           .= g.x by NDef;
        hence thesis;
      end;
      suppose
B1:     x > 0;
        f.x = I.(x,0) by FNeg
           .= max (1 - x, 0) by A1,B1,FUZIMPL1:def 23
           .= 1 - x by XXREAL_0:def 10,A2
           .= g.x by NDef;
        hence thesis;
      end;
    end;
    hence thesis by FUNCT_2:63;
  end;
