
theorem Th10:
  for L being RelStr, X,Y being set, x being Element of L holds (x
is_<=_than X & x is_<=_than Y implies x is_<=_than X \/ Y) & (x is_>=_than X &
  x is_>=_than Y implies x is_>=_than X \/ Y)
proof
  let L be RelStr, X,Y be set, x be Element of L;
  thus x is_<=_than X & x is_<=_than Y implies x is_<=_than X \/ Y
  proof
    assume
A1: ( for y being Element of L st y in X holds y >= x)& for y being
    Element of L st y in Y holds y >= x;
    let y be Element of L;
    y in X \/ Y implies y in X or y in Y by XBOOLE_0:def 3;
    hence thesis by A1;
  end;
  assume
A2: ( for y being Element of L st y in X holds y <= x)& for y being
  Element of L st y in Y holds y <= x;
  let y be Element of L;
  y in X \/ Y implies y in X or y in Y by XBOOLE_0:def 3;
  hence thesis by A2;
end;
