reserve A,B,p,q,r,s for Element of LTLB_WFF,
  n for Element of NAT,
  X for Subset of LTLB_WFF,
  g for Function of LTLB_WFF,BOOLEAN,
  x,y for set;

theorem Th4: for F, G being one-to-one FinSequence st rng F misses rng G
  holds F ^ G is one-to-one
  proof
    let F, G be one-to-one FinSequence;
    assume
A1: rng F misses rng G;
    reconsider FG = F ^ G as Function of (dom (F ^ G)),(rng (F ^ G))
    by FUNCT_2:1;
A2: FG is onto by FUNCT_2:def 3;
    card (dom (F ^ G)) = card Seg ((len F) + (len G)) by FINSEQ_1:def 7
    .= (len F) + (len G) by FINSEQ_1:57
    .= card (rng F) + (len G) by FINSEQ_4:62
    .= card (rng F) + card (rng G) by FINSEQ_4:62
    .= card ((rng F) \/ (rng G)) by A1,CARD_2:40
    .= card (rng (F ^ G)) by FINSEQ_1:31;
    hence F ^ G is one-to-one by A2,FINSEQ_4:63;
  end;
