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;

theorem
  for D1,D2 be non empty set, p be FinSequence of D1,
      f be Function of D1,D2 holds dom(f*p) = dom p &
  len (f*p) = len p &
  for n being Nat st n in dom (f*p) holds (f*p).n = f.(p.n)
proof
  let D1,D2 be non empty set, p be FinSequence of D1, f be Function of D1,D2;
A1: rng p c= D1 & dom f = D1 by FUNCT_2:def 1;
  hence dom(f*p) = dom p by RELAT_1:27;
  dom(f*p) = dom p by A1,RELAT_1:27;
  hence len(f*p) = len p by Th27;
  let n be Nat;
  assume n in dom (f*p);
  hence thesis by FUNCT_1:12;
end;
