
theorem
  for X, Y being non empty reflexive RelStr st [:X,Y:] is transitive
  holds X is transitive & Y is transitive
proof
  let X, Y be non empty reflexive RelStr such that
A1: [:X,Y:] is transitive;
  for x,y,z being Element of X st x <= y & y <= z holds x <= z
  proof
    set a = the Element of Y;
A2: a <= a;
    let x, y, z be Element of X;
    assume x <= y & y <= z;
    then [x,a] <= [y,a] & [y,a] <= [z,a] by A2,Th11;
    then [x,a] <= [z,a] by A1,YELLOW_0:def 2;
    hence thesis by Th11;
  end;
  hence X is transitive by YELLOW_0:def 2;
  for x,y,z being Element of Y st x <= y & y <= z holds x <= z
  proof
    set a = the Element of X;
A3: a <= a;
    let x, y, z be Element of Y;
    assume x <= y & y <= z;
    then [a,x] <= [a,y] & [a,y] <= [a,z] by A3,Th11;
    then [a,x] <= [a,z] by A1,YELLOW_0:def 2;
    hence thesis by Th11;
  end;
  hence thesis by YELLOW_0:def 2;
end;
