theorem Th40:
  v9 = v2 & v1 <> v2 implies Edges_Out(v9, X) = Edges_Out(v2, X)
proof
  assume that
A1: v9 = v2 and
A2: v1 <> v2;
  set G9 = AddNewEdge(v1, v2);
  set E = the carrier' of G;
  set S = the Source of G;
  set E9 = the carrier' of G9;
  set S9 = the Source of G9;
A3: E9 = E \/ {E} by Def7;
  now
    let x be object;
    hereby
      assume
A4:   x in Edges_Out(v9, X);
      then
A5:   x in X by Def2;
A6:   S9.x = v9 by A4,Def2;
      S9.E = v1 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 S.x = v2 by A1,A6,Th35;
      hence x in Edges_Out(v2, X) by A5,A7,Def2;
    end;
    assume
A8: x in Edges_Out(v2, X);
    then S.x = v2 by Def2;
    then
A9: S9.x = v9 by A1,A8,Th35;
    x in X & x in E9 by A3,A8,Def2,XBOOLE_0:def 3;
    hence x in Edges_Out(v9, X) by A9,Def2;
  end;
  hence thesis by TARSKI:2;
end;
