reserve G for Graph,
  v, v1, v2 for Vertex of G,
  c for Chain of G,
  p, p1, p2 for Path of G,
  vs, vs1, vs2 for FinSequence of the carrier of G,
  e, X for set,
  n, m for Nat;

theorem Th13:
  e in the carrier' of G & vs is_vertex_seq_of c & vs.len vs = (
the Target of G).e implies c^<*e*> is Chain of G & ex vs9 being FinSequence of
  the carrier of G st vs9 = vs^'<*(the Target of G).e, (the Source of G).e*> &
vs9 is_vertex_seq_of c^<*e*> & vs9.1 = vs.1 & vs9.len vs9 = (the Source of G).e
proof
  assume that
A1: e in the carrier' of G and
A2: vs is_vertex_seq_of c;
  reconsider ec = <*e*> as Chain of G by A1,MSSCYC_1:5;
  reconsider s = (the Source of G).e, t = (the Target of G).e as Vertex of G
  by A1,FUNCT_2:5;
  assume
A3: vs.len vs = (the Target of G).e;
  reconsider vse = <*t, s*> as FinSequence of the carrier of G;
A4: vse is_vertex_seq_of ec & vse.1 = t by Th11;
  hence c^<*e*> is Chain of G by A2,A3,GRAPH_2:43;
  reconsider ce = c^ec as Chain of G by A2,A3,A4,GRAPH_2:43;
  take vs9 = vs^'vse;
  thus vs9 = vs^'<*(the Target of G).e, (the Source of G).e*>;
  vs9 is_vertex_seq_of ce by A2,A3,A4,GRAPH_2:44;
  hence vs9 is_vertex_seq_of c^<*e*>;
  len vs = len c +1 by A2;
  then 1 <= len vs by NAT_1:11;
  hence vs9.1 = vs.1 by FINSEQ_6:140;
A5: len vse = 2 by FINSEQ_1:44;
  then vse.len vse = s;
  hence thesis by A5,FINSEQ_6:142;
end;
