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 Th34:
  (the carrier' of G) in the carrier' of AddNewEdge(v1, v2) & the
carrier' of G = (the carrier' of AddNewEdge(v1, v2)) \ {the carrier' of G} & (
  the Source of AddNewEdge(v1, v2)).(the carrier' of G) = v1 & (the Target of
  AddNewEdge(v1, v2)).(the carrier' of G) = v2
proof
  set G9 = AddNewEdge(v1, v2);
  set E = the carrier' of G;
  set S = the Source of G;
  set T = the Target of G;
  set E9 = the carrier' of G9;
A1: E9 = E \/ {E} by Def7;
  E in {E} by TARSKI:def 1;
  hence E in E9 by A1,XBOOLE_0:def 3;
  now
    let x be object;
    hereby
      assume
A2:   x in E; then
      reconsider x1=x as set;
      not x1 in x1;
      then x <> E by A2;
      then
A3:   not x in {E} by TARSKI:def 1;
      x in E9 by A1,A2,XBOOLE_0:def 3;
      hence x in E9 \ {E} by A3,XBOOLE_0:def 5;
    end;
    assume
A4: x in E9 \ {E};
    then not x in {E} by XBOOLE_0:def 5;
    hence x in E by A1,A4,XBOOLE_0:def 3;
  end;
  hence E = E9 \ {E} by TARSKI:2;
A5: E in dom (E .--> v1) by TARSKI:def 1;
  the Source of G9 = S +* (E .--> v1) by Def7;
  hence (the Source of G9).E = (E .--> v1).E by A5,FUNCT_4:13
    .= v1 by FUNCOP_1:72;
A6: E in dom (E .--> v2) by TARSKI:def 1;
  the Target of G9 = T +* (E .--> v2) by Def7;
  hence (the Target of G9).E = (E .--> v2).E by A6,FUNCT_4:13
    .= v2 by FUNCOP_1:72;
end;
