
theorem
  for t being t-conorm holds t <= drastic_conorm
  proof
    let t be t-conorm;
    set f1 = drastic_conorm;
    for a,b being Element of [.0,1.] holds
      f1.(a,b) >= t.(a,b)
    proof
      let a,b be Element of [.0,1.];
      per cases by XXREAL_0:15;
      suppose
A2:     a = 0; then
        reconsider aa = 0, bb = b as Element of [.0,1.];
        aa <= bb by XXREAL_1:1; then
        min (aa,bb) = 0 by XXREAL_0:def 9; then
A3:     f1.(aa,bb) = max(aa,bb) by Drastic2CDef;
        t.(aa,bb) = t.(bb,aa) by BINOP_1:def 2
                 .= b by ZeroDef;
        hence thesis by A2,A3,XXREAL_0:25;
      end;
      suppose
A2:     b = 0; then
        reconsider aa = a, bb = 0 as Element of [.0,1.];
        aa >= bb by XXREAL_1:1; then
        min (aa,bb) = 0 by XXREAL_0:def 9; then
A3:     f1.(aa,bb) = max(aa,bb) by Drastic2CDef;
        t.(aa,bb) = a by ZeroDef;
        hence thesis by A2,A3,XXREAL_0:25;
      end;
      suppose
aa:     min (a,b) <> 0;
        reconsider aa = a, bb = b as Element of [.0,1.];
        f1.(aa,bb) = 1 by Drastic2CDef,aa;
        hence thesis by XXREAL_1:1;
      end;
    end;
    hence thesis;
  end;
