reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;
reserve J for Nat;
reserve n for Nat;
reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;
reserve n for Nat;

theorem
  for p being n-element FinSequence,
      q being m-element FinSequence
   holds (p^q).(n+(1 qua Nat)) = q.1 & ... & (p^q).(n+m) = q.m
proof let p be n-element FinSequence, q be m-element FinSequence;
A1: len p = n by Th151;
A2: len q = m by Th151;
 let k be Nat;
 assume 1 <= k & k <= m;
  then
A3: n+(1 qua Nat) <= n+k & n+k <= n+m by XREAL_1:6;
 thus (p^q).(n+k)= (p^q).(n+k)
     .= q.(n+k-n) by A3,A1,A2,FINSEQ_1:23
     .= q.k;
end;
