
theorem Th12:
  for L being transitive RelStr for X being set, x,y being Element
  of L st X is_>=_than x & x >= y holds X is_>=_than y
proof
  let L be transitive RelStr;
  let X be set, x,y be Element of L such that
A1: for y being Element of L st y in X holds y >= x and
A2: x >= y;
  let z be Element of L;
  assume z in X;
  then z >= x by A1;
  hence thesis by A2,ORDERS_2:3;
end;
