reserve p, q for FinSequence,
  e,X for set,
  i, j, k, m, n for Nat,
  G for Graph;
reserve x,y,v,v1,v2,v3,v4 for Element of G;
reserve vs, vs1, vs2 for FinSequence of the carrier of G,
  c, c1, c2 for oriented Chain of G;
reserve sc for oriented simple Chain of G;

theorem Th18:
  for sc9 being oriented Chain of G st sc9=sc holds sc9 is Simple
proof
  let sc9 be oriented Chain of G;
  assume
A1: sc9=sc;
  consider vs such that
A2: vs is_oriented_vertex_seq_of sc9 by Th9;
  vs is_vertex_seq_of sc by A1,A2,Th4;
  then for n, m st 1<=n & n<m & m<=len vs & vs.n = vs.m holds n=1 & m=len vs
  by GRAPH_2:47;
  hence thesis by A2;
end;
