
theorem Th16:
  for S, T being with_suprema antisymmetric RelStr, x1, y1 being
Element of S, x2, y2 being Element of T holds [x1 "\/" y1, x2 "\/" y2] = [x1,x2
  ] "\/" [y1,y2]
proof
  let S, T be with_suprema antisymmetric RelStr, x1, y1 be Element of S, x2,
  y2 be Element of T;
A1: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
A2: ([x1,x2] "\/" [y1,y2])`2 = [x1,x2]`2 "\/" [y1,y2]`2 by Th14
    .= x2 "\/" [y1,y2]`2
    .= x2 "\/" y2
    .= [x1 "\/" y1, x2 "\/" y2]`2;
  ([x1,x2] "\/" [y1,y2])`1 = [x1,x2]`1 "\/" [y1,y2]`1 by Th14
    .= x1 "\/" [y1,y2]`1
    .= x1 "\/" y1
    .= [x1 "\/" y1, x2 "\/" y2]`1;
  hence thesis by A1,A2,DOMAIN_1:2;
end;
