reserve a,x,y for object, A,B for set,
  l,m,n for Nat;
reserve X,Y for set, x for object,
  p,q for Function-yielding FinSequence,
  f,g,h for Function;

theorem Th57:
  for p being FinSequence holds firstdom (<*f*>^p) = dom f &
  lastrng (p^<*f*>) = rng f
proof
  let p be FinSequence;
  thus firstdom (<*f*>^p) = proj1((<*f*>^p).1) by Def5
    .= dom f by FINSEQ_1:41;
  len <*f*> = 1 by FINSEQ_1:40;
  then len (p^<*f*>) = len p+1 by FINSEQ_1:22;
  hence lastrng (p^<*f*>) = proj2((p^<*f*>).(len p+1)) by Def6
    .= rng f by FINSEQ_1:42;
end;
