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;

theorem
  for p being n-element FinSequence, q being FinSequence
    holds (p^q).1 = p.1 & ... & (p^q).n = p.n
proof let p be n-element FinSequence, q be FinSequence;
   let k be Nat;
A1: len p = n by Th151;
   assume 1 <= k & k <= n;
   then k in dom p by A1,Th25;
   hence (p^q).k = p.k by FINSEQ_1:def 7;
end;
