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 Th26:
  e in the carrier' of G & not e in X & (v = (the Target of G).e
  or v = (the Source of G).e) implies Degree v <> Degree(v, X)
proof
  set T = the Target of G, S = the Source of G, E = the carrier' of G;
  assume that
A1: e in E and
A2: not e in X and
A3: v = T.e or v = S.e;
A4: Degree v = Degree(v, E) by Th24;
  Edges_Out v = Edges_Out(v, E);
  then
A5: Edges_Out(v, X) c= Edges_Out(v, E) by Th21;
  Edges_In v = Edges_In(v, E);
  then
A6: Edges_In(v, X) c= Edges_In(v, E) by Th20;
  per cases by A3;
  suppose
A7: v = T.e;
A8: not e in Edges_In(v, X) by A2,Def1;
    e in Edges_In(v, E) by A1,A7,Def1;
    then Edges_In(v, X) c< Edges_In(v, E) by A6,A8;
    then card Edges_In(v, X) < card Edges_In(v, E) by CARD_2:48;
    hence thesis by A4,A5,NAT_1:43,XREAL_1:8;
  end;
  suppose
A9: v = S.e;
A10: not e in Edges_Out(v, X) by A2,Def2;
    e in Edges_Out(v, E) by A1,A9,Def2;
    then Edges_Out(v, X) c< Edges_Out(v, E) by A5,A10;
    then card Edges_Out(v, X) < card Edges_Out(v, E) by CARD_2:48;
    hence thesis by A4,A6,NAT_1:43,XREAL_1:8;
  end;
end;
