
theorem
  for f,g being Fuzzy_Negation holds
    min(f,g) = (minfuncreal [.0,1.]).(f,g)
  proof
    let f,g be Fuzzy_Negation;
    set A = [.0,1.];
    set fg = min (f,g);
    set mfg = (minfuncreal [.0,1.]).(f,g);
B1: dom fg = [.0,1.] by FUNCT_2:def 1;
    f in Funcs(A,REAL) & g in Funcs(A,REAL) & Funcs (A,REAL) <> {}
      by LemmaFunc2; then
    (minfuncreal A).(f,g) in Funcs (A,REAL) by BINOP_1:17; then
    consider f1 being Function such that
C1: mfg = f1 & dom f1 = A & rng f1 c= REAL by FUNCT_2:def 2;
    for x being object st x in [.0,1.] holds
      fg.x = mfg.x
    proof
      let x be object;
      assume D0: x in [.0,1.];
      consider f2,g2 being Function of [.0,1.], REAL such that
D1:   f2 = f & g2 = g & fg = min (f2,g2) by MinFuz;
      dom f = A & rng f c= REAL &
      dom g = A & rng g c= REAL by FUNCT_2:def 1,RELAT_1:def 19; then
      reconsider fA = f, gA = g as Element of Funcs(A,REAL)
        by FUNCT_2:def 2;
      dom (minreal.:(fA,gA)) = A by FUNCT_2:def 1; then
      (minreal.:(fA,gA)).x = minreal.(fA.x,gA.x) by D0,FUNCOP_1:22
         .= min (fA.x,gA.x) by REAL_LAT:def 1
         .= fg.x by D1,D0,COUSIN2:def 1;
      hence thesis by REAL_LAT:def 5;
    end;
    hence thesis by C1,B1,FUNCT_1:2;
  end;
