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);

theorem Th43:
  v9 = v & v <> v2 implies Edges_In(v9, X) = Edges_In(v, X)
proof
  assume that
A1: v9 = v and
A2: v <> v2;
  set G9 = AddNewEdge(v1, v2);
  set E = the carrier' of G;
  set T = the Target of G;
  set E9 = the carrier' of G9;
  set T9 = the Target of G9;
A3: E9 = E \/ {E} by Def7;
  now
    let x be object;
    hereby
      assume
A4:   x in Edges_In(v9, X);
      then
A5:   x in X by Def1;
A6:   T9.x = v9 by A4,Def1;
      T9.E = v2 by Th34;
      then not x in {E} by A1,A2,A6,TARSKI:def 1;
      then
A7:   x in E by A3,A4,XBOOLE_0:def 3;
      then T.x = v by A1,A6,Th35;
      hence x in Edges_In(v, X) by A5,A7,Def1;
    end;
    assume
A8: x in Edges_In(v, X);
    then T.x = v by Def1;
    then
A9: T9.x = v9 by A1,A8,Th35;
    x in X & x in E9 by A3,A8,Def1,XBOOLE_0:def 3;
    hence x in Edges_In(v9, X) by A9,Def1;
  end;
  hence thesis by TARSKI:2;
end;
