
theorem Th22:
  for L being non empty transitive RelStr for x,y being Element of L st x <= y
  holds uparrow y c= uparrow x
proof
  let L be non empty transitive RelStr;
  let x,y be Element of L such that
A1: x <= y;
  let z be object;
  assume
A2: z in uparrow y;
  then reconsider z as Element of L;
  y <= z by A2,Th18;
  then x <= z by A1,ORDERS_2:3;
  hence thesis by Th18;
end;
