now :: thesis: ex GG being [ELabeled] inducedSubgraph of G,V,E st GG is elabel-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 ) ; :: thesis: ex GG being [ELabeled] inducedSubgraph of G,V,E st GG is elabel-inheriting

set X = the [ELabeled] inducedSubgraph of G,V,E;
set EL = (the_ELabel_of G) | (the_Edges_of the [ELabeled] inducedSubgraph of G,V,E);
reconsider EL9 = (the_ELabel_of G) | (the_Edges_of the [ELabeled] inducedSubgraph of G,V,E) as PartFunc of (dom ((the_ELabel_of G) | (the_Edges_of the [ELabeled] inducedSubgraph of G,V,E))),(rng ((the_ELabel_of G) | (the_Edges_of the [ELabeled] inducedSubgraph of G,V,E))) by RELSET_1:4;
reconsider EL9 = EL9 as PartFunc of (the_Edges_of the [ELabeled] inducedSubgraph of G,V,E),(rng ((the_ELabel_of G) | (the_Edges_of the [ELabeled] inducedSubgraph of G,V,E))) by RELAT_1:58, RELSET_1:5;
set GG = the [ELabeled] inducedSubgraph of G,V,E .set (ELabelSelector,EL9);
A2: the [ELabeled] inducedSubgraph of G,V,E .set (ELabelSelector,EL9) == the [ELabeled] inducedSubgraph of G,V,E by Lm3;
then the [ELabeled] inducedSubgraph of G,V,E .set (ELabelSelector,EL9) is Subgraph of the [ELabeled] inducedSubgraph of G,V,E by GLIB_000:87;
then reconsider GG = the [ELabeled] inducedSubgraph of G,V,E .set (ELabelSelector,EL9) as Subgraph of G by GLIB_000:43;
A3: the_Vertices_of GG = the_Vertices_of the [ELabeled] inducedSubgraph of G,V,E by A2
.= V by A1, GLIB_000:def 37 ;
the_Edges_of GG = the_Edges_of the [ELabeled] inducedSubgraph of G,V,E by A2
.= E by A1, GLIB_000:def 37 ;
then reconsider GG = GG as [ELabeled] inducedSubgraph of G,V,E by A1, A3, GLIB_000:def 37;
take GG = GG; :: thesis: GG is elabel-inheriting
the_ELabel_of GG = (the_ELabel_of G) | (the_Edges_of the [ELabeled] inducedSubgraph of G,V,E) by GLIB_000:8
.= (the_ELabel_of G) | (the_Edges_of GG) by A2 ;
hence GG is elabel-inheriting ; :: thesis: verum

end;

suppose A4: ( not V is non empty Subset of (the_Vertices_of G) or not E c= G .edgesBetween V ) ; :: thesis: ex GG being [ELabeled] inducedSubgraph of G,V,E st GG is elabel-inheriting

reconsider GG = G as Subgraph of G by GLIB_000:40;
reconsider GG = GG as [ELabeled] inducedSubgraph of G,V,E by A4, GLIB_000:def 37;
take GG = GG; :: thesis: GG is elabel-inheriting
dom (the_ELabel_of G) c= the_Edges_of G by Lm1;
then the_ELabel_of G = (the_ELabel_of G) | (the_Edges_of G) by RELAT_1:68;
hence GG is elabel-inheriting ; :: thesis: verum

end;

end;

end;

hence ex b₁ being [ELabeled] inducedSubgraph of G,V,E st b₁ is elabel-inheriting ; :: thesis: verum