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 Th56:
  for G be finite connected Graph, c be Element of G-CycleSet st
rng c <> the carrier' of G & c is non empty holds {v9 where v9 is Vertex of G :
v9 in G-VSet rng c & Degree v9 <> Degree(v9, rng c)} is non empty Subset of the
  carrier of G
proof
  let G be finite connected Graph, c be Element of G-CycleSet;
  defpred P[Vertex of G] means $1 in G-VSet rng c & Degree $1 <> Degree($1,
  rng c);
  set X = {v9 where v9 is Vertex of G : P[v9]};
  set T = the Target of G;
  set S = the Source of G;
  set E = the carrier' of G;
A1: rng c c= E by FINSEQ_1:def 4;
  reconsider cp = c as cyclic Path of G by Def8;
  assume that
A2: rng c <> the carrier' of G and
A3: c is non empty;
  consider vs being FinSequence of the carrier of G such that
A4: vs is_vertex_seq_of cp by GRAPH_2:33;
A5: G-VSet rng cp = rng vs by A3,A4,GRAPH_2:31;
  now
    consider x being object such that
A6: not (x in rng c iff x in E) by A2,TARSKI:2;
    reconsider x as Element of E by A1,A6;
    reconsider v = (the Target of G).x as Vertex of G by A1,A6,FUNCT_2:5;
    per cases;
    suppose
A7:   v in rng vs;
      Degree v <> Degree(v, rng c) by A1,A6,Th26;
      hence
      ex v being Vertex of G st v in rng vs & Degree v <> Degree(v, rng c
      ) by A7;
    end;
    suppose
A8:   not v in rng vs;
A9:   ex e being object st e in rng c by A3,XBOOLE_0:def 1;
      then rng c meets E by A1,XBOOLE_0:3;
      then consider v9 being Vertex of G, e being Element of E such that
A10:  v9 in rng vs and
A11:  ( not e in rng c)&( v9 = T.e or v9 = S.e) by A5,A8,Th19;
      Degree v9 <> Degree(v9, rng c) by A1,A9,A11,Th26;
      hence
      ex v being Vertex of G st v in rng vs & Degree v <> Degree(v, rng c
      ) by A10;
    end;
  end;
  then consider v being Vertex of G such that
A12: v in rng vs & Degree v <> Degree(v, rng c);
A13: X is Subset of the carrier of G from DOMAIN_1:sch 7;
  v in X by A5,A12;
  hence thesis by A13;
end;
