theorem Th67:
  for F being Function st [:rng p,{a}:] c= dom F & r = F[:](p,a)
  holds len r = len p
proof
  let F be Function;
  assume [:rng p,{a}:] c= dom F;
  then dom(F[:](p,a)) = dom p by Lm5;
  then dom(F[:](p,a)) = Seg len p by FINSEQ_1:def 3;
  hence thesis by FINSEQ_1:def 3;
end;
