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 Th15:
  for p being Simple oriented Chain of G holds p is one-to-one
proof
  let p be Simple oriented Chain of G;
  set VV=the carrier of G;
  consider vs being FinSequence of VV such that
A1: vs is_oriented_vertex_seq_of p 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;
A3: len vs = len p + 1 by A1,GRAPH_4:def 5;
  now
    let n,m;
    assume that
A4: 1<=n and
A5: n<m and
A6: m<=len p;
A7: 1 <= m by A4,A5,XXREAL_0:2;
    then
A8: p.m orientedly_joins vs/.m, vs/.(m+1) by A1,A6,GRAPH_4:def 5;
    assume
A9: p.n = p.m;
A10: n <= len p by A5,A6,XXREAL_0:2;
    then p.n orientedly_joins vs/.n, vs/.(n+1) by A1,A4,GRAPH_4:def 5;
    then
A11: vs/.n = (the Source of G).(p.m) by A9,GRAPH_4:def 1
      .= vs/.m by A8,GRAPH_4:def 1;
A12: len p < len vs by A3,XREAL_1:29;
    then n <= len vs by A10,XXREAL_0:2;
    then n in dom vs by A4,FINSEQ_3:25;
    then
A13: vs.n=vs/.n by PARTFUN1:def 6;
A14: m <= len vs by A6,A12,XXREAL_0:2;
    then m in dom vs by A7,FINSEQ_3:25;
    then vs.m=vs.n by A11,A13,PARTFUN1:def 6;
    then m= len vs by A2,A4,A5,A14;
    hence contradiction by A3,A6,XREAL_1:29;
  end;
  hence thesis by Th4;
end;
