
theorem Th6:
  for L being RelStr holds L is transitive iff L opp is transitive
proof
  let L be RelStr;
  thus L is transitive implies L opp is transitive
  proof
    assume
A1: L is transitive;
    let x,y,z be Element of L opp;
    assume x <= y & z >= y;
    then ~x >= ~y & ~z <= ~y by Th1;
    hence thesis by Th1,A1;
  end;
  assume
A2: L opp is transitive;
  let x,y,z be Element of L;
  assume x <= y & z >= y;
  then x~ >= y~ & z~ <= y~ by LATTICE3:9;
  then x~ >= z~ by A2;
  hence thesis by LATTICE3:9;
end;
