
theorem
  for L being complete non empty Poset for A, B being non empty Subset
  of L holds sup (A "\/" B) = sup A "\/" sup B
proof
  let L be complete non empty Poset, A, B be non empty Subset of L;
  B is_<=_than sup (A "\/" B) by Th25;
  then
A1: sup B <= sup (A "\/" B) by YELLOW_0:32;
  A is_<=_than sup (A "\/" B) by Th25;
  then sup A <= sup (A "\/" B) by YELLOW_0:32;
  then
A2: sup A "\/" sup B <= sup (A "\/" B) "\/" sup (A "\/" B) by A1,YELLOW_3:3;
  A is_<=_than sup A & B is_<=_than sup B by YELLOW_0:32;
  then ex_sup_of A "\/" B,L & A "\/" B is_<=_than sup A "\/" sup B by Th27,
YELLOW_0:17;
  then
A3: sup (A "\/" B) <= sup A "\/" sup B by YELLOW_0:def 9;
  sup (A "\/" B) <= sup (A "\/" B);
  then sup (A "\/" B) "\/" sup (A "\/" B) = sup (A "\/" B) by YELLOW_0:24;
  hence thesis by A3,A2,ORDERS_2:2;
end;
