
theorem
  for t being t-norm holds
    t <= minnorm
  proof
    let t be t-norm;
    set f1 = minnorm;
    for a,b being Element of [.0,1.] holds
      t.(a,b) <= f1.(a,b)
    proof
      let a,b be Element of [.0,1.];
      reconsider aa = a, bb = b as Element of [.0,1.];
A1:   f1.(a,b) = min (aa,bb) by MinDef;
      reconsider cc = 1 as Element of [.0,1.] by XXREAL_1:1;
      aa <= 1 by XXREAL_1:1; then
      t.(aa,bb) <= t.(cc,bb) by MonDef; then
      t.(aa,bb) <= t.(bb,cc) by BINOP_1:def 2; then
A3:   t.(aa,bb) <= bb by IdDef;
      bb <= 1 by XXREAL_1:1; then
      t.(aa,bb) <= t.(aa,cc) by MonDef; then
      t.(aa,bb) <= aa by IdDef;
      hence thesis by A1,XXREAL_0:20,A3;
    end;
    hence thesis;
  end;
