reserve D for non empty set,
  f for FinSequence of D,
  p, p1, p2, p3, q for Element of D,
  i, j, k, l, n for Nat;

theorem
  i <> k & j <> k & l <> i & l <> j & 1 <= i & i <= len f & 1 <= j & j
<= len f & 1 <= k & k <= len f & 1 <= l & l <= len f implies Swap(Swap(f, i, j)
  , k, l) = Swap(Swap(f, k, l), i, j)
proof
  assume that
A1: i <> k & j <> k and
A2: l <> i & l <> j and
A3: 1 <= i and
A4: i <= len f and
A5: 1 <= j and
A6: j <= len f and
A7: 1 <= k and
A8: k <= len f and
A9: 1 <= l and
A10: l <= len f;
A11: i <= len Replace(f,k,f/.l) & j <= len Replace(f,k,f/.l)
     by A4,A6,FUNCT_7:97;
  set F = Swap(f,i,j);
  k <= len F & l <= len F by A8,A10,Th18;
  then Swap(F,k,l) = Replace(Replace(F,k,F/.l),l,F/.k) by A7,A9,Def2
    .= Replace(Replace(F,k,F/.l),l,f/.k) by A1,A7,A8,Th30
    .= Replace(Replace(F,k,f/.l),l,f/.k) by A2,A9,A10,Th30
    .= Replace(Replace(Swap(f,j,i),k,f/.l),l,f/.k) by Th21
    .= Replace(Swap(Replace(f,k,f/.l),j,i),l,f/.k) by A1,A3,A4,A5,A6,Th33
    .= Swap(Replace(Replace(f,k,f/.l),l,f/.k),j,i) by A2,A3,A5,A11,Th33
    .= Swap(Swap(f,k,l),j,i) by A7,A8,A9,A10,Def2
    .= Swap(Swap(f,k,l),i,j) by Th21;
  hence thesis;
end;
