
theorem
  for S, T being non empty RelStr, x, y, z being Element of [:S,T:]
holds x is_>=_than {y,z} iff x`1 is_>=_than {y`1,z`1} & x`2 is_>=_than {y`2,z`2
  }
proof
  let S, T be non empty RelStr, x, y, z be Element of [:S,T:];
  thus x is_>=_than {y,z} implies x`1 is_>=_than {y`1,z`1} & x`2 is_>=_than {y
  `2,z`2}
  proof
    assume
A1: for b being Element of [:S,T:] st b in {y,z} holds x >= b;
A2: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
    then y = [y`1,y`2] by MCART_1:21;
    then [y`1,y`2] in {y,z} by TARSKI:def 2;
    then
A3: x >= [y`1,y`2] by A1;
    z = [z`1,z`2] by A2,MCART_1:21;
    then [z`1,z`2] in {y,z} by TARSKI:def 2;
    then
A4: x >= [z`1,z`2] by A1;
A5: x = [x`1,x`2] by A2,MCART_1:21;
    hereby
      let b be Element of S;
      assume b in {y`1,z`1};
      then b = y`1 or b = z`1 by TARSKI:def 2;
      hence x`1 >= b by A3,A4,A5,YELLOW_3:11;
    end;
    let b be Element of T;
    assume b in {y`2,z`2};
    then b = y`2 or b = z`2 by TARSKI:def 2;
    hence thesis by A3,A4,A5,YELLOW_3:11;
  end;
  assume that
A6: for b being Element of S st b in {y`1,z`1} holds x`1 >= b and
A7: for b being Element of T st b in {y`2,z`2} holds x`2 >= b;
  let b be Element of [:S,T:];
  assume b in {y,z};
  then
A8: b = y or b = z by TARSKI:def 2;
  then b`2 in {y`2,z`2} by TARSKI:def 2;
  then
A9: x`2 >= b`2 by A7;
  b`1 in {y`1,z`1} by A8,TARSKI:def 2;
  then x`1 >= b`1 by A6;
  hence thesis by A9,YELLOW_3:12;
end;
