reserve n,m,i,j,k for Nat,
  x,y,e,X,V,U for set,
  W,f,g for Function;
reserve p,q for FinSequence;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G;
reserve v,v1,v2,v3 for Element of G;
reserve p,q for oriented Chain of G;
reserve G for finite Graph,
  ps for Simple oriented Chain of G,
  P,Q for oriented Chain of G,
  v1,v2,v3 for Element of G,
  pe,qe for FinSequence of the carrier' of G;

theorem
  len ps <= VerticesCount G
proof
  set VV=the carrier of G;
  consider vs being FinSequence of VV such that
A1: vs is_oriented_vertex_seq_of ps and
A2: for n,m st 1<=n & n<m & m<=len vs & vs.n=vs.m holds n=1 & m=len vs
  by GRAPH_4:def 7;
  reconsider V =VV as finite set;
A3: len vs = len ps + 1 by A1,GRAPH_4:def 5;
  then vs <> {};
  then consider q being FinSequence, x be object such that
A4: vs = q^<*x*> by FINSEQ_1:46;
A5: len ps + 1 = len q + len <*x*> by A3,A4,FINSEQ_1:22
    .=len q + 1 by FINSEQ_1:39;
  now
    let n,m;
    assume that
A6: 1<=n and
A7: n<m and
A8: m<=len q;
    1 <= m by A6,A7,XXREAL_0:2;
    then
A9: m in dom q by A8,FINSEQ_3:25;
    n <= len q by A7,A8,XXREAL_0:2;
    then n in dom q by A6,FINSEQ_3:25;
    then
A10: vs.n=q.n by A4,FINSEQ_1:def 7;
    len q < len vs by A3,A5,XREAL_1:29;
    then
A11: m < len vs by A8,XXREAL_0:2;
    assume q.n = q.m;
    then vs.m=vs.n by A4,A10,A9,FINSEQ_1:def 7;
    hence contradiction by A2,A6,A7,A11;
  end;
  then
A12: card(rng q)=len q by Th5;
  rng vs c= VV & rng q c= rng vs by A4,FINSEQ_1:29,def 4;
  then card(rng q) <= card V by NAT_1:43,XBOOLE_1:1;
  hence thesis by A5,A12,GRAPH_1:def 19;
end;
