
theorem Th14:
  for S, T being with_suprema antisymmetric RelStr, x, y being
Element of [:S,T:] holds (x "\/" y)`1 = x`1 "\/" y`1 & (x "\/" y)`2 = x`2 "\/"
  y`2
proof
  let S, T be with_suprema antisymmetric RelStr, x, y be Element of [:S,T:];
  set a = (x "\/" y)`1, b = (x "\/" y)`2;
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;
A3: y = [y`1,y`2] by A1,MCART_1:21;
A4: for d being Element of S st d >= x`1 & d >= y`1 holds a <= d
  proof
    set t = x`2 "\/" y`2;
    let d be Element of S such that
A5: d >= x`1 and
A6: d >= y`1;
    t >= y`2 by YELLOW_0:22;
    then
A7: [d,t] >= y by A3,A6,YELLOW_3:11;
    t >= x`2 by YELLOW_0:22;
    then [d,t] >= x by A2,A5,YELLOW_3:11;
    then
A8: x "\/" y <= [d,t] by A7,YELLOW_0:22;
    [d,t]`1 = d;
    hence thesis by A8,YELLOW_3:12;
  end;
A9: for d being Element of T st d >= x`2 & d >= y`2 holds b <= d
  proof
    set s = x`1 "\/" y`1;
    let d be Element of T such that
A10: d >= x`2 and
A11: d >= y`2;
    s >= y`1 by YELLOW_0:22;
    then
A12: [s,d] >= y by A3,A11,YELLOW_3:11;
    s >= x`1 by YELLOW_0:22;
    then [s,d] >= x by A2,A10,YELLOW_3:11;
    then
A13: x "\/" y <= [s,d] by A12,YELLOW_0:22;
    [s,d]`2 = d;
    hence thesis by A13,YELLOW_3:12;
  end;
  x "\/" y >= y by YELLOW_0:22;
  then
A14: a >= y`1 & b >= y`2 by YELLOW_3:12;
  x "\/" y >= x by YELLOW_0:22;
  then a >= x`1 & b >= x`2 by YELLOW_3:12;
  hence thesis by A14,A4,A9,YELLOW_0:18;
end;
