
theorem Th12: :: transitivity
  for X being set, P, Q, S being Dependency of X st P <= Q & Q <=
  S holds P <= S
proof
  let X be set, p, q, r be Dependency of X;
  assume that
A1: p <= q and
A2: q <= r;
A3: q`2 c= r`2 by A2;
  p`2 c= q`2 by A1;
  then
A4: p`2 c= r`2 by A3;
A5: r`1 c= q`1 by A2;
  q`1 c= p`1 by A1;
  then r`1 c= p`1 by A5;
  hence thesis by A4;
end;
