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;
reserve G for finite Graph,
  v for Vertex of G,
  c for Chain of G,
  vs for FinSequence of the carrier of G,
  X1, X2 for set;
reserve G for Graph,
  v, v1, v2 for Vertex of G,
  c for Chain of G,
  p for Path of G,
  vs for FinSequence of the carrier of G,
  v9 for Vertex of AddNewEdge(v1, v2),
  p9 for Path of AddNewEdge(v1, v2),
  vs9 for FinSequence of the carrier of AddNewEdge(v1, v2);
reserve G for finite Graph,
  v, v1, v2 for Vertex of G,
  vs for FinSequence of the carrier of G,
  v9 for Vertex of AddNewEdge(v1, v2);
reserve G for Graph,
  v for Vertex of G,
  vs for FinSequence of the carrier of G;
reserve G for finite Graph,
  v for Vertex of G,
  vs for FinSequence of the carrier of G;

theorem Th58:
  for G being connected Graph, p being Path of G, vs being
FinSequence of the carrier of G st p is Eulerian & vs is_vertex_seq_of p holds
  rng vs = the carrier of G
proof
  let G be connected Graph, p be Path of G, vs be FinSequence of the carrier
  of G such that
A1: p is Eulerian and
A2: vs is_vertex_seq_of p and
A3: rng vs <> the carrier of G;
  consider x being object such that
A4: x in rng vs & not x in the carrier of G or x in the carrier of G &
  not x in rng vs by A3,TARSKI:2;
  vs <> {} by A2;
  then consider y being object such that
A5: y in rng vs by XBOOLE_0:def 1;
A6: rng vs c= the carrier of G by FINSEQ_1:def 4;
  then consider
  c being Chain of G, vs1 being FinSequence of the carrier of G such
  that
A7: c is non empty and
A8: vs1 is_vertex_seq_of c & vs1.1 = x and
  vs1.len vs1 = y by A4,A5,Th18;
A9: 1 <= len c by A7,NAT_1:14;
A10: rng p = the carrier' of G by A1;
  reconsider c as FinSequence of the carrier' of G by MSSCYC_1:def 1;
  1 in dom c by A9,FINSEQ_3:25;
  then c.1 in the carrier' of G by PARTFUN1:4;
  then (the Target of G).(c.1) in rng vs & (the Source of G).(c.1) in rng vs
  by A2,A10,Th15;
  hence contradiction by A6,A4,A8,A9,Lm3;
end;
