let X be set ; :: thesis:
let G be Graph; :: thesis:
let v1, v2 be Vertex of G; :: thesis:
let v9 be Vertex of (AddNewEdge (v1,v2)); :: thesis:
assume that
A1: v9 = v2 and
A2: the carrier' of G in X ; :: thesis:
set G9 = AddNewEdge (v1,v2);
set E = the carrier' of G;
set T = the Target of G;
set E9 = the carrier' of (AddNewEdge (v1,v2));
set T9 = the Target of (AddNewEdge (v1,v2));
A3: the carrier' of (AddNewEdge (v1,v2)) = the carrier' of G \/ { the carrier' of G} by Def7;

let x be object ; :: thesis:

hereby :: thesis:

assume A4: x in Edges_In (v9,X) ; :: thesis:
then A5: x in X by Def1;
A6: the Target of (AddNewEdge (v1,v2)) . x = v9 by A4, Def1;

per cases by A3, A4, XBOOLE_0:def 3;

suppose A7: x in the carrier' of G ; :: thesis:

then the Target of G . x = v2 by A1, A6, Th35;
then x in Edges_In (v2,X) by A5, A7, Def1;
hence x in (Edges_In (v2,X)) \/ { the carrier' of G} by XBOOLE_0:def 3; :: thesis:

end;

suppose x in { the carrier' of G} ; :: thesis:

hence x in (Edges_In (v2,X)) \/ { the carrier' of G} by XBOOLE_0:def 3; :: thesis:

end;

end;

end;

assume A8: x in (Edges_In (v2,X)) \/ { the carrier' of G} ; :: thesis:

per cases by A8, XBOOLE_0:def 3;

suppose A9: x in Edges_In (v2,X) ; :: thesis:

then the Target of G . x = v2 by Def1;
then A10: the Target of (AddNewEdge (v1,v2)) . x = v9 by A1, A9, Th35;
( x in X & x in the carrier' of (AddNewEdge (v1,v2)) ) by A3, A9, Def1, XBOOLE_0:def 3;
hence x in Edges_In (v9,X) by A10, Def1; :: thesis:

end;

suppose A11: x in { the carrier' of G} ; :: thesis:

A12: the Target of (AddNewEdge (v1,v2)) . the carrier' of G = v2 by Th34;
( x = the carrier' of G & x in the carrier' of (AddNewEdge (v1,v2)) ) by A3, A11, TARSKI:def 1, XBOOLE_0:def 3;
hence x in Edges_In (v9,X) by A1, A2, A12, Def1; :: thesis:

end;

end;

end;

hence Edges_In (v9,X) = (Edges_In (v2,X)) \/ { the carrier' of G} by TARSKI:2; :: thesis:
assume (Edges_In (v2,X)) /\ { the carrier' of G} <> {} ; :: according to XBOOLE_0:def 7 :: thesis:
then consider x being object such that
A13: x in (Edges_In (v2,X)) /\ { the carrier' of G} by XBOOLE_0:def 1;
x in { the carrier' of G} by A13, XBOOLE_0:def 4;
then A14: x = the carrier' of G by TARSKI:def 1;
A: x in the carrier' of G by A13;
reconsider xx = x as set by TARSKI:1;
not xx in xx ;
hence contradiction by A14, A; :: thesis: