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;

theorem Th11:
  {} is Simple oriented Chain of G
proof
  set v = the Element of G;
  set vs=<*v*>;
A1: now
    let n,m;
    assume that
A2: 1<=n & n<m & m<=len vs and
    vs.n=vs.m;
    assume not (n=1 & m=len vs);
    len vs =1 by FINSEQ_1:39;
    hence contradiction by A2,XXREAL_0:2;
  end;
  vs is_oriented_vertex_seq_of {} qua FinSequence by GRAPH_4:8;
  hence thesis by A1,GRAPH_1:14,GRAPH_4:def 7;
end;
