
theorem
  for t being t-norm holds
    drastic_norm <= t
  proof
    let t be t-norm;
    set f1 = drastic_norm;
    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;
      suppose
A2:     a = 1;
        reconsider aa = 1, bb = b as Element of [.0,1.] by XXREAL_1:1;
        t.(aa,bb) = t.(bb,aa) by BINOP_1:def 2
                 .= b by IdDef;
        hence thesis by DrasticDef,A2;
      end;
      suppose
A2:     b = 1;
        reconsider aa = a, bb = 1 as Element of [.0,1.] by XXREAL_1:1;
        f1.(aa,bb) = aa by DrasticDef;
        hence thesis by A2,IdDef;
      end;
      suppose
A2:     a <> 1 & b <> 1;
        reconsider aa = a, bb = b as Element of [.0,1.];
        f1.(aa,bb) = 0 by DrasticDef,A2;
        hence thesis by XXREAL_1:1;
      end;
    end;
    hence thesis;
  end;
