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;
