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;
