
theorem
  for L being antisymmetric transitive RelStr, X,Y being set st
  ex_inf_of X,L & ex_inf_of Y,L & ex_inf_of X \/ Y, L holds "/\"(X \/ Y, L) =
  "/\"(X,L) "/\" "/\"(Y,L)
proof
  let L be antisymmetric transitive RelStr;
  let X,Y be set such that
A1: ex_inf_of X,L & ex_inf_of Y,L and
A2: ex_inf_of X \/ Y, L;
  set a = "/\"(X,L), b = "/\"(Y,L), c = "/\"(X \/ Y, L);
A3: a is_<=_than X & b is_<=_than Y by A1,Th31;
A4: now
    let d be Element of L;
    assume d <= a & d <= b;
    then d is_<=_than X & d is_<=_than Y by A3,Th4;
    then d is_<=_than X \/ Y by Th10;
    hence c >= d by A2,Th31;
  end;
  c <= a & c <= b by A1,A2,Th35,XBOOLE_1:7;
  hence thesis by A4,Th19;
end;
