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;
