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;
