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;

theorem Th33:
  for G being non void finite connected Graph, v being Vertex of G
  holds Degree v <> 0
proof
  let G be non void finite connected Graph, v be Vertex of G;
  assume
A1: Degree v = 0;
  set E = the carrier' of G;
A2: Degree v = Degree(v, E) by Th24
    .= card Edges_In(v, E) + card Edges_Out(v, E);
  then
A3: Edges_In(v, E) = {} by A1;
A4: Edges_Out(v, E) = {} by A1,A2;
  set S = the Source of G;
  set T = the Target of G;
  consider e being object such that
A5: e in E by XBOOLE_0:def 1;
  reconsider s = S.e as Vertex of G by A5,FUNCT_2:5;
  per cases;
  suppose
    v = s;
    hence contradiction by A4,A5,Def2;
  end;
  suppose
    v <> s;
    then consider
    c be Chain of G, vs be FinSequence of the carrier of G such that
A6: c is non empty and
A7: vs is_vertex_seq_of c and
A8: vs.1 = v and
    vs.len vs = s by Th18;
A9: 0+1 <= len c by A6,NAT_1:13;
    then 1 in dom c by FINSEQ_3:25;
    then
A10: c.1 in rng c by FUNCT_1:def 3;
    c is FinSequence of the carrier' of G by MSSCYC_1:def 1;
    then
A11: rng c c= the carrier' of G by FINSEQ_1:def 4;
    vs.1 = T.(c.1) or vs.1 = S.(c.1) by A7,A9,Lm3;
    hence contradiction by A3,A4,A8,A11,A10,Def1,Def2;
  end;
end;
