now ex GG being [Weighted] inducedSubgraph of G,V,E st GG is weight-inheriting per cases
( ( V is non empty Subset of (the_Vertices_of G) & E c= G .edgesBetween V ) or not V is non empty Subset of (the_Vertices_of G) or not E c= G .edgesBetween V )
;
suppose A1:
(
V is non
empty Subset of
(the_Vertices_of G) &
E c= G .edgesBetween V )
;
ex GG being [Weighted] inducedSubgraph of G,V,E st GG is weight-inheriting set X = the
[Weighted] inducedSubgraph of
G,
V,
E;
set W =
(the_Weight_of G) | (the_Edges_of the [Weighted] inducedSubgraph of G,V,E);
dom (the_Weight_of G) = the_Edges_of G
by PARTFUN1:def 2;
then
dom ((the_Weight_of G) | (the_Edges_of the [Weighted] inducedSubgraph of G,V,E)) = the_Edges_of the
[Weighted] inducedSubgraph of
G,
V,
E
by RELAT_1:62;
then reconsider W =
(the_Weight_of G) | (the_Edges_of the [Weighted] inducedSubgraph of G,V,E) as
ManySortedSet of
the_Edges_of the
[Weighted] inducedSubgraph of
G,
V,
E by PARTFUN1:def 2;
set GG = the
[Weighted] inducedSubgraph of
G,
V,
E .set (
WeightSelector,
W);
A2:
the
[Weighted] inducedSubgraph of
G,
V,
E .set (
WeightSelector,
W)
== the
[Weighted] inducedSubgraph of
G,
V,
E
by Lm3;
then
the
[Weighted] inducedSubgraph of
G,
V,
E .set (
WeightSelector,
W) is
Subgraph of the
[Weighted] inducedSubgraph of
G,
V,
E
by GLIB_000:87;
then reconsider GG = the
[Weighted] inducedSubgraph of
G,
V,
E .set (
WeightSelector,
W) as
Subgraph of
G by GLIB_000:43;
A3:
the_Vertices_of GG =
the_Vertices_of the
[Weighted] inducedSubgraph of
G,
V,
E
by A2
.=
V
by A1, GLIB_000:def 37
;
the_Edges_of GG =
the_Edges_of the
[Weighted] inducedSubgraph of
G,
V,
E
by A2
.=
E
by A1, GLIB_000:def 37
;
then reconsider GG =
GG as
[Weighted] inducedSubgraph of
G,
V,
E by A1, A3, GLIB_000:def 37;
take GG =
GG;
GG is weight-inheriting the_Weight_of GG =
W
by GLIB_000:8
.=
(the_Weight_of G) | (the_Edges_of GG)
by A2
;
hence
GG is
weight-inheriting
;
verum end;
hence
ex b1 being [Weighted] inducedSubgraph of G,V,E st b1 is weight-inheriting
; verum