
theorem CoHam:
  conorm Hamacher_norm = Hamacher_conorm
  proof
    set dn = conorm Hamacher_norm;
    set dc = Hamacher_conorm;
    for a,b being Element of [.0,1.] holds
      dc.(a,b) = 1 - (Hamacher_norm).(1-a,1-b)
    proof
      let a,b be Element of [.0,1.];
AB:   0 in [.0,1.] by XXREAL_1:1;
A3:   1 - a in [.0,1.] & 1 - b in [.0,1.] by OpIn01;
WE:   0 <= a <= 1 & 0 <= b <= 1 by XXREAL_1:1;
      per cases;
      suppose
Aa:     a <> 1 or b <> 1; then
AA:     1 - a * b <> 0 by WE,XREAL_1:150;
        dc.(a,b) = (a + b - 2 * a * b) / (1 - a * b) by HamCoDef,Aa
        .= ((1 * ((1 - a) + (1 - b) - (1 - a) * (1 - b))) -
           ((1 - a) * (1 - b)))
           / ((1 - a) + (1 - b) - (1 - a) * (1 - b))
        .= 1 - ((1 - a) * (1 - b)) / ((1 - a) + (1 - b) - (1 - a) * (1 - b))
           by XCMPLX_1:127,AA
        .= 1 - (Hamacher_norm).(1-a,1-b) by HamDef,A3;
        hence thesis;
      end;
      suppose
T1:     a = b = 1; then
        Hamacher_norm.(1-a,1-b) = (0 * 0) / (0 + 0 - 0 * 0) by HamDef,AB
          .= 0;
        hence thesis by T1,HamCoDef;
      end;
    end;
    hence thesis by CoDef;
  end;
