
theorem
  FNegation I_YG = NegationD1
  proof
    set I = I_YG;
    set f = FNegation I;
    set g = NegationD1;
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.];
      per cases;
      suppose
        x <= 0; then
B2:     x = 0 by XXREAL_1:1;
        f.x = I.(x,0) by FNeg
           .= 1 by FUZIMPL1:def 21,B2
           .= g.x by D1Def,B2;
        hence thesis;
      end;
      suppose
B1:     x > 0;
        f.x = I.(x,0) by FNeg
           .= 0 to_power x by A1,B1,FUZIMPL1:def 21
           .= 0 by B1,POWER:def 2
           .= g.x by D1Def,B1;
        hence thesis;
      end;
    end;
    hence thesis by FUNCT_2:63;
  end;
