theorem Th39:
  v9 = v1 & v1 <> v2 implies Edges_In(v9, X) = Edges_In(v1, X)
proof
  assume that
A1: v9 = v1 and
A2: v1 <> v2;
  set G9 = AddNewEdge(v1, v2);
  set E = the carrier' of G;
  set T = the Target of G;
  set E9 = the carrier' of G9;
  set T9 = the Target of G9;
A3: E9 = E \/ {E} by Def7;
  now
    let x be object;
    hereby
      assume
A4:   x in Edges_In(v9, X);
      then
A5:   x in X by Def1;
A6:   T9.x = v9 by A4,Def1;
      T9.E = v2 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 T.x = v1 by A1,A6,Th35;
      hence x in Edges_In(v1, X) by A5,A7,Def1;
    end;
    assume
A8: x in Edges_In(v1, X);
    then T.x = v1 by Def1;
    then
A9: T9.x = v9 by A1,A8,Th35;
    x in X & x in E9 by A3,A8,Def1,XBOOLE_0:def 3;
    hence x in Edges_In(v9, X) by A9,Def1;
  end;
  hence thesis by TARSKI:2;
end;
