
theorem NormIs0:
  for t being commutative monotonic with-1-identity BinOp of [.0,1.] holds
    for a being Element of [.0,1.] holds
      t.(a,0) = 0
  proof
    let t be commutative monotonic with-1-identity BinOp of [.0,1.];
    let a be Element of [.0,1.];
T0: 0 in [.0,1.] & 1 in [.0,1.] by XXREAL_1:1; then
T3: t.(1,0) = t.(0,1) by BINOP_1:def 2 .= 0 by T0,IdDef;
    for a being Element of [.0,1.] holds t.(a,0) = 0
    proof
      let a be Element of [.0,1.];
      t.(a,0) in [.0,1.] by NormIn01; then
T4:   0 <= t.(a,0) by XXREAL_1:1;
      a <= 1 by XXREAL_1:1;
      hence thesis by T4,T0,MonDef,T3;
    end;
    hence thesis;
  end;
