reserve G for Graph,
  k, m, n for Nat;
reserve G for non void Graph;

theorem Th6:
  for G being Graph, e being set st e in the carrier' of G for fe
being directed Chain of G st fe = <*e*> holds vertex-seq fe = <*(the Source of
  G).e, (the Target of G).e*>
proof
  let G be Graph;
  let e be set;
  assume e in the carrier' of G;
  then reconsider
  so = (the Source of G).e, ta = (the Target of G).e as Element of
  the carrier of G by FUNCT_2:5;
  reconsider sota = <*so, ta*> as FinSequence of the carrier of G;
  let fe be directed Chain of G;
  assume
A1: fe = <*e*>;
  then
A2: len fe = 1 by FINSEQ_1:39;
A3: sota is_vertex_seq_of fe
  proof
    thus len sota = len fe + 1 by A2,FINSEQ_1:44;
    let n;
A4: sota/.2=ta by FINSEQ_4:17;
    assume 1<=n & n<=len fe;
    then
A5: n=1 by A2,XXREAL_0:1;
    e joins so, ta & sota/.1 = so by FINSEQ_4:17;
    hence thesis by A1,A5,A4;
  end;
  e = fe.1 by A1;
  then sota.1 = (the Source of G).(fe.1);
  hence thesis by A1,A3,GRAPH_2:def 10;
end;
