theorem Th42:
  v9 = v2 & (the carrier' of G) in X implies Edges_In(v9, X) =
  Edges_In(v2, X) \/ {the carrier' of G} & Edges_In(v2, X) misses {the carrier'
  of G}
proof
  assume that
A1: v9 = v2 and
A2: (the carrier' of G) in X;
  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;
      per cases by A3,A4,XBOOLE_0:def 3;
      suppose
A7:     x in E;
        then T.x = v2 by A1,A6,Th35;
        then x in Edges_In(v2, X) by A5,A7,Def1;
        hence x in Edges_In(v2, X) \/ {E} by XBOOLE_0:def 3;
      end;
      suppose
        x in {E};
        hence x in Edges_In(v2, X) \/ {E} by XBOOLE_0:def 3;
      end;
    end;
    assume
A8: x in Edges_In(v2, X) \/ {E};
    per cases by A8,XBOOLE_0:def 3;
    suppose
A9:   x in Edges_In(v2, X);
      then T.x = v2 by Def1;
      then
A10:  T9.x = v9 by A1,A9,Th35;
      x in X & x in E9 by A3,A9,Def1,XBOOLE_0:def 3;
      hence x in Edges_In(v9, X) by A10,Def1;
    end;
    suppose
A11:  x in {E};
A12:  T9.E = v2 by Th34;
      x = E & x in E9 by A3,A11,TARSKI:def 1,XBOOLE_0:def 3;
      hence x in Edges_In(v9, X) by A1,A2,A12,Def1;
    end;
  end;
  hence Edges_In(v9, X) = Edges_In(v2, X) \/ {E} by TARSKI:2;
  assume Edges_In(v2, X) /\ {E} <> {};
  then consider x being object such that
A13: x in Edges_In(v2, X) /\ {E} by XBOOLE_0:def 1;
  x in {E} by A13,XBOOLE_0:def 4;
  then
A14: x = E by TARSKI:def 1;
A: x in E by A13;
  reconsider xx = x as set by TARSKI:1;
  not xx in xx;
  hence contradiction by A14,A;
end;
