reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;

theorem Th65:
  for F being Function st [:{a},rng p:] c= dom F & r = F[;](a,p)
  holds len r = len p
proof
  let F be Function;
  assume [:{a},rng p:] c= dom F;
  then dom(F[;](a,p)) = dom p by Lm4;
  then dom(F[;](a,p)) = Seg len p by FINSEQ_1:def 3;
  hence thesis by FINSEQ_1:def 3;
end;
