
theorem :: Proposition 1.4.17 iii)
  for I being satisfying_(EP) satisfying_(OP) BinOp of [.0,1.] holds
    (FNegation I) * (FNegation I) * (FNegation I) = FNegation I
  proof
    let I be satisfying_(EP) satisfying_(OP) BinOp of [.0,1.];
    set f = FNegation I;
    now let x be Element of [.0,1.];
      f is non-increasing by N2Def; then
R2:   f.(f.(f.x)) <= f.x by NonInc,Prop1417ii;
v1:   f.x = I.(x,0) by FNeg;
v2:   I.(I.(I.(x,0),0),0) = I.(I.(f.x,0),0) by FNeg
           .= I.(f.(f.x),0) by FNeg
           .= f.(f.(f.x)) by FNeg;
vz:   f.(f.x) = I.(I.(x,0),0) by FNeg,v1;
r1:   dom (f * f) = [.0,1.] by FUNCT_2:def 1;
      0 in [.0,1.] by XXREAL_1:1; then
      I.(I.(x,0),I.(I.(I.(x,0),0),0)) = I.(I.(I.(x,0),0),I.(I.(x,0),0))
             by FUZIMPL2:def 2,vz,v1
           .= 1 by vz,FUZIMPL2:def 3; then
      f.x <= I.(I.(f.x,0),0) by v1,v2,FUZIMPL2:def 4; then
      f.x <= I.(f.(f.x),0) by FNeg; then
      f.x <= f.(f.(f.x)) by FNeg; then
      f.x = f.(f.(f.x)) by R2,XXREAL_0:1; then
      f.x = f.((f * f).x) by r1,FUNCT_1:12
         .= (f * (f * f)).x by FUNCT_1:13,r1;
      hence (f * f * f).x = f.x by RELAT_1:36;
    end;
    hence thesis by FUNCT_2:63;
  end;
