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 Th19:
  for sc9 being Simple oriented Chain of G
  holds sc9 is oriented simple Chain of G
proof
  let sc9 be Simple oriented Chain of G;
  consider vs such that
A1: vs is_oriented_vertex_seq_of sc9 and
A2: for n,m st 1<=n & n<m & m<=len vs & vs.n=vs.m holds n=1 & m=len vs by Def7;
  vs is_vertex_seq_of sc9 by A1,Th4;
  hence thesis by A2,GRAPH_2:def 9;
end;
