reserve x,y,X for set,
  i,j,k,m,n for Nat,
  p for FinSequence of X,
  ii for Integer;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G,
  p,q for oriented Chain of G,
  W for Function,
  U,V,e,ee for set,
  v1,v2,v3,v4 for Vertex of G;
reserve G for finite Graph,
  P,Q for oriented Chain of G,
  v1,v2,v3 for Vertex of G;
reserve G for finite oriented Graph,
  P,Q for oriented Chain of G,
  W for Function of (the carrier' of G), Real>=0,
  v1,v2,v3,v4 for Vertex of G;
reserve f,g,h for Element of REAL*,
  r for Real;
reserve G for oriented Graph,
  v1,v2 for Vertex of G,
  W for Function of (the carrier' of G), Real>=0;

theorem
  for G being oriented Graph,vs be FinSequence,p,q being oriented Chain
  of G st p is_oriented_edge_seq_of vs & q is_oriented_edge_seq_of vs holds p=q
proof
  let G be oriented Graph,vs be FinSequence,p,q be oriented Chain of G;
  assume that
A1: p is_oriented_edge_seq_of vs and
A2: q is_oriented_edge_seq_of vs;
A3: len p+1 = len vs by A1
    .= len q +1 by A2;
  now
    let k be Nat;
    assume
A4: 1<=k & k <= len p;
    then
A5: (the Target of G).(p.k) = vs.(k+1) by A1
      .=(the Target of G).(q.k) by A2,A3,A4;
A6: p.k in the carrier' of G & q.k in the carrier' of G by A3,A4,Th2;
    (the Source of G).(p.k) = vs.k by A1,A4
      .=(the Source of G).(q.k) by A2,A3,A4;
    hence p.k=q.k by A5,A6,GRAPH_1:def 7;
  end;
  hence thesis by A3,FINSEQ_1:14;
end;
