reserve i,j,k,n,m for Nat,
        b,b1,b2 for bag of n;

theorem Th1:
  for F, G being XFinSequence st F^G is one-to-one holds
    rng F misses rng G
proof
  let F, G be XFinSequence such that
A1: F^G is one-to-one;
  assume rng F meets rng G;
  then consider y be object such that
A2: y in rng F & y in rng G by XBOOLE_0:3;
 consider n1 be object such that
A3: n1 in dom F & F.n1 = y by A2,FUNCT_1:def 3;
 consider n2 be object such that
A4: n2 in dom G & G.n2 = y by A2,FUNCT_1:def 3;
  reconsider n1,n2 as Nat by A3,A4;
A5: (F^G).n1 = F.n1 & (F^G).(len F+n2) = G.n2 by A3,A4,AFINSQ_1:def 3;
  dom F c= dom (F^G) by AFINSQ_1:21;then
  n1 in dom (F^G) & (len F+n2) in dom (F^G) by A3,A4,AFINSQ_1:23;
  then n1 = n2+len F & n1 < len F by A5,A1,FUNCT_1:def 4,A3,A4,AFINSQ_1:86;
  hence thesis by NAT_1:11;
end;
