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 Th65:
  for q being non-empty non empty FinSequence, p being
  FuncSequence of q st p <> {} holds firstdom p = q.1 & lastrng p c= q.len q
proof
  let q be non-empty non empty FinSequence;
  let p be FuncSequence of q;
  assume
A1: p <> {};
  then 1 in dom p by FINSEQ_5:6;
  then p.1 in Funcs(q.1,q.(1+1)) by Def9;
  then ex f being Function st p.1 = f & dom f = q.1 & rng f c= q.2 by
FUNCT_2:def 2;
  hence firstdom p = q.1 by A1,Def5;
  len p in dom p by A1,FINSEQ_5:6;
  then p.len p in Funcs(q.len p,q.(len p+1)) by Def9;
  then
A2: ex g being Function st p.len p = g & dom g = q.len p & rng g c= q.(len p+
  1) by FUNCT_2:def 2;
  len p+1 = len q by Def9;
  hence thesis by A1,A2,Def6;
end;
