let X be set ; for G being Graph
for v2, v, v1 being Vertex of G
for v' being Vertex of (AddNewEdge v1,v2) st v' = v & v <> v1 holds
Edges_Out v',X = Edges_Out v,X
let G be Graph; for v2, v, v1 being Vertex of G
for v' being Vertex of (AddNewEdge v1,v2) st v' = v & v <> v1 holds
Edges_Out v',X = Edges_Out v,X
let v2, v, v1 be Vertex of G; for v' being Vertex of (AddNewEdge v1,v2) st v' = v & v <> v1 holds
Edges_Out v',X = Edges_Out v,X
let v' be Vertex of (AddNewEdge v1,v2); ( v' = v & v <> v1 implies Edges_Out v',X = Edges_Out v,X )
assume that
A1:
v' = v
and
A2:
v <> v1
; Edges_Out v',X = Edges_Out v,X
set G' = AddNewEdge v1,v2;
set E = the carrier' of G;
set S = the Source of G;
set E' = the carrier' of (AddNewEdge v1,v2);
set S' = the Source of (AddNewEdge v1,v2);
A3:
the carrier' of (AddNewEdge v1,v2) = the carrier' of G \/ {the carrier' of G}
by Def7;
now let x be
set ;
( ( x in Edges_Out v',X implies x in Edges_Out v,X ) & ( x in Edges_Out v,X implies x in Edges_Out v',X ) )hereby ( x in Edges_Out v,X implies x in Edges_Out v',X )
assume A4:
x in Edges_Out v',
X
;
x in Edges_Out v,Xthen A5:
x in X
by Def2;
A6:
the
Source of
(AddNewEdge v1,v2) . x = v'
by A4, Def2;
the
Source of
(AddNewEdge v1,v2) . the
carrier' of
G = v1
by Th39;
then
not
x in {the carrier' of G}
by A1, A2, A6, TARSKI:def 1;
then A7:
x in the
carrier' of
G
by A3, A4, XBOOLE_0:def 3;
then
the
Source of
G . x = v
by A1, A6, Th40;
hence
x in Edges_Out v,
X
by A5, A7, Def2;
verum
end; assume A8:
x in Edges_Out v,
X
;
x in Edges_Out v',Xthen
the
Source of
G . x = v
by Def2;
then A9:
the
Source of
(AddNewEdge v1,v2) . x = v'
by A1, A8, Th40;
(
x in X &
x in the
carrier' of
(AddNewEdge v1,v2) )
by A3, A8, Def2, XBOOLE_0:def 3;
hence
x in Edges_Out v',
X
by A9, Def2;
verum end;
hence
Edges_Out v',X = Edges_Out v,X
by TARSKI:2; verum