
theorem Th4:
  for L being transitive RelStr, x,y being Element of L st x <= y
for X being set holds (y is_<=_than X implies x is_<=_than X) & (x is_>=_than X
  implies y is_>=_than X)
proof
  let L be transitive RelStr, x,y be Element of L;
  assume
A1: x <= y;
  let X be set;
  hereby
    assume
A2: y is_<=_than X;
    thus x is_<=_than X
    proof
      let z be Element of L;
      assume z in X;
      then z >= y by A2;
      hence thesis by A1,ORDERS_2:3;
    end;
  end;
  assume
A3: x is_>=_than X;
  let z be Element of L;
  assume z in X;
  then x >= z by A3;
  hence thesis by A1,ORDERS_2:3;
end;
