
theorem LEM1:
  for X being non empty set,
      R being Relation of X holds
    R /\ (R~)` is asymmetric
  proof
    let X be non empty set,
        R be Relation of X;
    set P = R /\ (R~)`;
    assume not P is asymmetric; then
    consider x, y being object such that
A1: [x,y] in P & [y,x] in P by LemAsym;
    [x,y] in R & [x,y] in (R~)` by A1,XBOOLE_0:def 4; then
    not [x,y] in R~ by XBOOLE_0:def 5; then
    not [y,x] in R by RELAT_1:def 7;
    hence thesis by A1,XBOOLE_0:def 4;
  end;
