
theorem DiffMax:
  for a,b being Element of [.0,1.] st max (a,b) <> 1 holds
    a <> 1 & b <> 1
  proof
    let a,b be Element of [.0,1.];
    assume
A1: max (a,b) <> 1;
A2: a <= 1 & b <= 1 by XXREAL_1:1;
    assume a = 1 or b = 1; then
    per cases;
    suppose a = 1;
      hence thesis by A1,A2,XXREAL_0:def 10;
    end;
    suppose b = 1;
      hence thesis by A1,A2,XXREAL_0:def 10;
    end;
  end;
