reserve UA for Universal_Algebra,
  f, g for Function of UA, UA;
reserve I for set,
  A, B, C for ManySortedSet of I;

theorem Th14:
  for F, G being Function-yielding Function st F is "1-1" & G is
  "1-1" holds G ** F is "1-1"
proof
  let F, G be Function-yielding Function such that
A1: F is "1-1" and
A2: G is "1-1";
  let i be set, f be Function such that
A3: i in dom (G**F) and
A4: (G**F).i = f;
A5: dom (G**F) = (dom G) /\ (dom F) by PBOOLE:def 19;
  then i in dom F by A3,XBOOLE_0:def 4;
  then
A6: F.i is one-to-one by A1;
  i in dom G by A3,A5,XBOOLE_0:def 4;
  then G.i is one-to-one by A2;
  then (G.i)*(F.i) is one-to-one by A6;
  hence thesis by A3,A4,PBOOLE:def 19;
end;
