reserve x, x1, x2, y, z, X9 for set,
  X, Y for finite set,
  n, k, m for Nat,
  f for Function;
reserve F,Ch for Function;
reserve Fy for finite-yielding Function;

theorem Th51:
  for F, G be XFinSequence st F is one-to-one & G is one-to-one &
  rng F misses rng G holds F^G is one-to-one
proof
  let F, G be XFinSequence such that
A1: F is one-to-one and
A2: G is one-to-one and
A3: rng F misses rng G;
  len F,rng F are_equipotent by A1,WELLORD2:def 4;
  then
A4: card len F=card rng F by CARD_1:5;
  len G,rng G are_equipotent by A2,WELLORD2:def 4;
  then
A5: card len G=card rng G by CARD_1:5;
  reconsider FG=F^G as Function of dom (F^G),rng (F^G) by FUNCT_2:1;
A6: dom (F^G)=len F+len G by AFINSQ_1:def 3;
A7: FG is onto by FUNCT_2:def 3;
  card(rng F\/rng G)=card rng F + card rng G by A3,CARD_2:40;
  then card dom (F^G)=card rng (F^G) by A4,A5,A6,AFINSQ_1:26;
  hence thesis by A7,FINSEQ_4:63;
end;
