
theorem
  for S, T being with_suprema antisymmetric RelStr, X, Y being Subset of
  [:S,T:] holds proj1 (X "\/" Y) = proj1 X "\/" proj1 Y & proj2 (X "\/" Y) =
  proj2 X "\/" proj2 Y
proof
  let S, T be with_suprema antisymmetric RelStr, X, Y be Subset of [:S,T:];
A1: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
A2: X "\/" Y = { x "\/" y where x, y is Element of [:S,T:]: x in X & y in Y
  } by YELLOW_4:def 3;
A3: proj1 X "\/" proj1 Y = { x "\/" y where x, y is Element of S: x in proj1
  X & y in proj1 Y } by YELLOW_4:def 3;
  hereby
    hereby
      let a be object;
      assume a in proj1 (X "\/" Y);
      then consider b being object such that
A4:   [a,b] in X "\/" Y by XTUPLE_0:def 12;
      consider x, y being Element of [:S,T:] such that
A5:   [a,b] = x "\/" y and
A6:   x in X and
A7:   y in Y by A2,A4;
      x = [x`1,x`2] by A1,MCART_1:21;
      then
A8:   x`1 in proj1 X by A6,XTUPLE_0:def 12;
      y = [y`1,y`2] by A1,MCART_1:21;
      then
A9:   y`1 in proj1 Y by A7,XTUPLE_0:def 12;
      a = [a,b]`1
        .= x`1 "\/" y`1 by A5,Th14;
      hence a in proj1 X "\/" proj1 Y by A8,A9,YELLOW_4:10;
    end;
    let a be object;
    assume a in proj1 X "\/" proj1 Y;
    then consider x, y being Element of S such that
A10: a = x "\/" y and
A11: x in proj1 X and
A12: y in proj1 Y by A3;
    consider x2 being object such that
A13: [x,x2] in X by A11,XTUPLE_0:def 12;
    consider y2 being object such that
A14: [y,y2] in Y by A12,XTUPLE_0:def 12;
    reconsider x2, y2 as Element of T by A1,A13,A14,ZFMISC_1:87;
    [x,x2] "\/" [y,y2] = [a,x2 "\/" y2] by A10,Th16;
    then [a,x2 "\/" y2] in X "\/" Y by A13,A14,YELLOW_4:10;
    hence a in proj1 (X "\/" Y) by XTUPLE_0:def 12;
  end;
  hereby
    let b be object;
    assume b in proj2 (X "\/" Y);
    then consider a being object such that
A15: [a,b] in X "\/" Y by XTUPLE_0:def 13;
    consider x, y being Element of [:S,T:] such that
A16: [a,b] = x "\/" y and
A17: x in X and
A18: y in Y by A2,A15;
    x = [x`1,x`2] by A1,MCART_1:21;
    then
A19: x`2 in proj2 X by A17,XTUPLE_0:def 13;
    y = [y`1,y`2] by A1,MCART_1:21;
    then
A20: y`2 in proj2 Y by A18,XTUPLE_0:def 13;
    b = [a,b]`2
      .= x`2 "\/" y`2 by A16,Th14;
    hence b in proj2 X "\/" proj2 Y by A19,A20,YELLOW_4:10;
  end;
  let b be object;
A21: proj2 X "\/" proj2 Y = { x "\/" y where x, y is Element of T: x in
  proj2 X & y in proj2 Y } by YELLOW_4:def 3;
  assume b in proj2 X "\/" proj2 Y;
  then consider x, y being Element of T such that
A22: b = x "\/" y and
A23: x in proj2 X and
A24: y in proj2 Y by A21;
  consider x1 being object such that
A25: [x1,x] in X by A23,XTUPLE_0:def 13;
  consider y1 being object such that
A26: [y1,y] in Y by A24,XTUPLE_0:def 13;
  reconsider x1, y1 as Element of S by A1,A25,A26,ZFMISC_1:87;
  [x1,x] "\/" [y1,y] = [x1 "\/" y1,b] by A22,Th16;
  then [x1 "\/" y1,b] in X "\/" Y by A25,A26,YELLOW_4:10;
  hence thesis by XTUPLE_0:def 13;
end;
