
theorem
  for S, T being non empty RelStr, x, y being Element of [:S,T:] holds x
  is_<=_than {y} iff x`1 is_<=_than {y`1} & x`2 is_<=_than {y`2}
proof
  let S, T be non empty RelStr, x, y be Element of [:S,T:];
  thus x is_<=_than {y} implies x`1 is_<=_than {y`1} & x`2 is_<=_than {y`2}
  proof
A1: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
    then
A2: x = [x`1,x`2] by MCART_1:21;
    y = [y`1,y`2] by A1,MCART_1:21;
    then
A3: [y`1,y`2] in {y} by TARSKI:def 1;
    assume for b being Element of [:S,T:] st b in {y} holds x <= b;
    then
A4: x <= [y`1,y`2] by A3;
    hereby
      let b be Element of S;
      assume b in {y`1};
      then b = y`1 by TARSKI:def 1;
      hence x`1 <= b by A4,A2,YELLOW_3:11;
    end;
    let b be Element of T;
    assume b in {y`2};
    then b = y`2 by TARSKI:def 1;
    hence thesis by A4,A2,YELLOW_3:11;
  end;
  assume that
A5: for b being Element of S st b in {y`1} holds x`1 <= b and
A6: for b being Element of T st b in {y`2} holds x`2 <= b;
  let b be Element of [:S,T:];
  assume b in {y};
  then
A7: b = y by TARSKI:def 1;
  then b`2 in {y`2} by TARSKI:def 1;
  then
A8: x`2 <= b`2 by A6;
  b`1 in {y`1} by A7,TARSKI:def 1;
  then x`1 <= b`1 by A5;
  hence thesis by A8,YELLOW_3:12;
end;
