theorem
  q in AcyclicPaths(p) implies len q <= len p
proof
A1: card (rng p) <= card (dom p) by CARD_2:47;
A2: card (dom p) = card (Seg len p) by FINSEQ_1:def 3
    .= len p by FINSEQ_1:57;
  assume q in AcyclicPaths(p);
  then consider x being Simple oriented Chain of G such that
A3: q=x and
  x <> {} and
  (the Source of G).(x.1) = (the Source of G).(p.1) and
  (the Target of G).(x.(len x)) = (the Target of G).(p.(len p)) and
A4: rng (x) c= rng p;
  x is one-to-one by Th15;
  then
A5: card(rng x)=len x by FINSEQ_4:62;
  card (rng x) <= card (rng p) by A4,NAT_1:43;
  hence thesis by A3,A5,A1,A2,XXREAL_0:2;
end;
