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 Th35:
  e in the carrier' of G implies (the Source of AddNewEdge(v1, v2)
).e = (the Source of G).e & (the Target of AddNewEdge(v1, v2)).e = (the Target
  of G).e
proof
  set S = the Source of G;
  set T = the Target of G;
  set E = the carrier' of G;
  set G9 = AddNewEdge(v1, v2);
  set S9 = the Source of G9;
  set T9 = the Target of G9;
  assume
A1: e in the carrier' of G;
A2: not e in dom (E .--> v1)
  proof
    assume e in dom(E .--> v1);
    then e in {E};
    then e = E by TARSKI:def 1;
    hence contradiction by A1;
  end;
  thus S9.e = (S +* (E .--> v1)).e by Def7
    .= S.e by A2,FUNCT_4:11;
A3: not e in dom (E .--> v2)
  proof
    assume e in dom(E .--> v2);
    then e in {E};
    then e = E by TARSKI:def 1;
    hence contradiction by A1;
  end;
  thus T9.e = (T +* (E .--> v2)).e by Def7
    .= T.e by A3,FUNCT_4:11;
end;
