reserve x,y for set;
reserve G for Graph;
reserve vs,vs9 for FinSequence of the carrier of G;
reserve IT for oriented Chain of G;
reserve N for Nat;
reserve n,m,k,i,j for Nat;
reserve r,r1,r2 for Real;

theorem Th1:
  for vs st IT is Simple & vs is_oriented_vertex_seq_of IT holds
  for n,m st 1<=n & n<m & m<=len vs & vs.n=vs.m holds n=1 & m=len vs
proof
  let vs;
  assume that
A1: IT is Simple and
A2: vs is_oriented_vertex_seq_of IT;
A3: len vs = len IT + 1 by A2,GRAPH_4:def 5;
  consider vs9 such that
A4: vs9 is_oriented_vertex_seq_of IT and
A5: for n,m being Nat
   st 1<=n & n<m & m<=len vs9 & vs9.n=vs9.m holds n=1 & m=len vs9
    by A1,GRAPH_4:def 7;
A6: for n,m being Nat
   st 1<=n & n<m & m<=len vs9 & vs9.n=vs9.m holds n=1 & m=len vs9
   by A5;
  per cases;
  suppose
    IT<> {};
    then vs=vs9 by A2,A4,GRAPH_4:10;
    hence thesis by A6;
  end;
  suppose
    IT= {};
    then len IT=0;
    hence thesis by A3,XXREAL_0:2;
  end;
end;
