reserve x for set,
  i,j,k,n for Nat,
  K for Field;
reserve a,b,c,d for Element of K;
reserve D for non empty set;

theorem Th17:
  for f, g being FinSequence st f ^ g in Permutations n holds f ^
  Rev g in Permutations n
proof
  let f, g be FinSequence;
A1: rng g = rng Rev g by FINSEQ_5:57;
  set h = f ^ Rev g;
  assume f ^ g in Permutations n;
  then
A2: f ^ g is Permutation of Seg n by MATRIX_1:def 12;
  then
A3: g is one-to-one by FINSEQ_3:91;
  dom (f ^ g) = Seg n by A2,FUNCT_2:52;
  then
A4: Seg n = Seg (len f + len g) by FINSEQ_1:def 7;
  len g = len Rev g by FINSEQ_5:def 3;
  then
A5: dom h = Seg n by A4,FINSEQ_1:def 7;
A6: rng (f^g) = rng f \/ rng g by FINSEQ_1:31
    .= rng (f^Rev g) by A1,FINSEQ_1:31;
A7: rng (f^g) = Seg n by A2,FUNCT_2:def 3;
  then reconsider h as FinSequence-like Function of Seg n, Seg n by A6,A5,
FUNCT_2:2;
A8: h is onto by A7,A6,FUNCT_2:def 3;
  rng f misses rng g & f is one-to-one by A2,FINSEQ_3:91;
  then h is one-to-one by A1,A3,FINSEQ_3:91;
  hence thesis by A8,MATRIX_1:def 12;
end;
