
theorem Th33:
  for L be non empty RelStr for S1,S2 be join-closed Subset of L
  holds S1 /\ S2 is join-closed
proof
  let L be non empty RelStr;
  let S1,S2 be join-closed Subset of L;
A1: subrelstr S2 is join-inheriting by Def2;
A2: subrelstr S1 is join-inheriting by Def2;
  now
    let x,y be Element of L;
    assume that
A3: x in the carrier of subrelstr (S1 /\ S2) and
A4: y in the carrier of subrelstr (S1 /\ S2) and
A5: ex_sup_of {x,y},L;
A6: y in S1 /\ S2 by A4,YELLOW_0:def 15;
    then y in S2 by XBOOLE_0:def 4;
    then
A7: y in the carrier of subrelstr S2 by YELLOW_0:def 15;
A8: x in S1 /\ S2 by A3,YELLOW_0:def 15;
    then x in S2 by XBOOLE_0:def 4;
    then x in the carrier of subrelstr S2 by YELLOW_0:def 15;
    then sup {x,y} in the carrier of subrelstr S2 by A1,A5,A7;
    then
A9: sup {x,y} in S2 by YELLOW_0:def 15;
    y in S1 by A6,XBOOLE_0:def 4;
    then
A10: y in the carrier of subrelstr S1 by YELLOW_0:def 15;
    x in S1 by A8,XBOOLE_0:def 4;
    then x in the carrier of subrelstr S1 by YELLOW_0:def 15;
    then sup {x,y} in the carrier of subrelstr S1 by A2,A5,A10;
    then sup {x,y} in S1 by YELLOW_0:def 15;
    then sup {x,y} in S1 /\ S2 by A9,XBOOLE_0:def 4;
    hence sup {x,y} in the carrier of subrelstr (S1 /\ S2) by YELLOW_0:def 15;
  end;
  then subrelstr (S1 /\ S2) is join-inheriting;
  hence thesis;
end;
