
theorem
  for I being Fuzzy_Implication,
      f being bijective increasing UnOp of [.0,1.] holds
    ConjImpl (I,f) = f" * I * [:f,f:]
  proof
    let I be Fuzzy_Implication,
        f be bijective increasing UnOp of [.0,1.];
    set g = ConjImpl (I,f);
    dom g = [:[.0,1.],[.0,1.]:] by FUNCT_2:def 1; then
A1: dom g = dom (f" * I * [:f,f:]) by FUNCT_2:def 1;
C2: dom f = [.0,1.] by FUNCT_2:def 1;
    for x being object st x in dom g holds
      g.x = (f" * I * [:f,f:]).x
    proof
      let x be object;
      assume x in dom g; then
W1:   x in [:[.0,1.],[.0,1.]:] by FUNCT_2:def 1; then
      consider x1, x2 being object such that
C0:   x1 in [.0,1.] & x2 in [.0,1.] & x = [x1,x2] by ZFMISC_1:def 2;
      reconsider x1, x2 as Element of [.0,1.] by C0;
D1:   x in dom [:f,f:] by W1,FUNCT_2:def 1;
X2:   dom I = [:[.0,1.],[.0,1.]:] by FUNCT_2:def 1;
      [f.x1,f.x2] in [:[.0,1.],[.0,1.]:] by ZFMISC_1:87; then
      [:f,f:].(x1,x2) in dom I by X2; then
D2:   [:f,f:].x in dom I by C0,BINOP_1:def 1;
      g.x = g.(x1,x2) by C0,BINOP_1:def 1
         .= f".(I.(f.x1, f.x2)) by BIDef
         .= f".(I. [f.x1, f.x2]) by BINOP_1:def 1
         .= f". (I.([:f,f:].(x1,x2))) by C2,FUNCT_3:def 8
         .= f". (I.([:f,f:].x)) by C0,BINOP_1:def 1
         .= (f" * I).(([:f,f:]).x) by FUNCT_1:13,D2
         .= (f" * I * [:f,f:]).x by FUNCT_1:13,D1;
      hence thesis;
    end;
    hence thesis by A1,FUNCT_1:2;
  end;
