theorem Th63:
  for F being Function st [:rng p,rng q:] c= dom F & r = F.:(p,q)
  holds len r = min(len p,len q)
proof
  let F be Function;
  reconsider k = min(len p,len q) as Element of NAT by XXREAL_0:15;
  assume [:rng p,rng q:] c= dom F;
  then dom(F.:(p,q)) = Seg k by Lm3;
  hence thesis by FINSEQ_1:def 3;
end;
