
theorem ConormNilmin:
  conorm nilmin_norm = nilmax_conorm
  proof
    set dn = conorm nilmin_norm;
    set dc = nilmax_conorm;
    for a,b being Element of [.0,1.] holds
      dc.(a,b) = 1 - (nilmin_norm).(1-a,1-b)
    proof
      let a,b be Element of [.0,1.];
A3:   1 - a in [.0,1.] & 1 - b in [.0,1.] by OpIn01;
      per cases;
      suppose
W0:     a + b < 1; then
        2 - (a + b) > 2 - 1 by XREAL_1:10; then
W1:     1 - a + (1 - b) > 1;
        dc.(a,b) = max (a,b) by NilmaxDef,W0
             .= 1 - min (1-a,1-b) by MaxMin
             .= 1 - (nilmin_norm).(1-a,1-b) by NilminDef,A3,W1;
        hence thesis;
      end;
      suppose
W0:     a + b >= 1; then
        2 - (a + b) <= 2 - 1 by XREAL_1:10; then
W1:     1 - a + (1 - b) <= 1;
        dc.(a,b) = 1 - 0 by NilmaxDef,W0
             .= 1 - (nilmin_norm).(1-a,1-b) by NilminDef,A3,W1;
        hence thesis;
      end;
    end;
    hence thesis by CoDef;
  end;
