let G1 be simple _Graph; :: thesis: for G2 being _Graph holds

( G1 == G2 iff G2 is SimpleGraph of G1 )

let G2 be _Graph; :: thesis: ( G1 == G2 iff G2 is SimpleGraph of G1 )

then consider E being RepEdgeSelection of G1 such that

A3: G2 is inducedSubgraph of G1, the_Vertices_of G1,E \ (G1 .loops()) by Def9;

the_Edges_of G1 = E \ (G1 .loops()) by Th74;

then G2 is inducedSubgraph of G1,(the_Vertices_of G1) by A3, GLIB_000:34;

hence G1 == G2 by GLIB_000:94; :: thesis: verum

( G1 == G2 iff G2 is SimpleGraph of G1 )

let G2 be _Graph; :: thesis: ( G1 == G2 iff G2 is SimpleGraph of G1 )

hereby :: thesis: ( G2 is SimpleGraph of G1 implies G1 == G2 )

assume
G2 is SimpleGraph of G1
; :: thesis: G1 == G2assume A1:
G1 == G2
; :: thesis: G2 is SimpleGraph of G1

set E = the RepEdgeSelection of G1;

A2: the_Edges_of G1 = the RepEdgeSelection of G1 \ (G1 .loops()) by Th74;

G1 is inducedSubgraph of G1,(the_Vertices_of G1) by GLIB_006:15;

then G1 is inducedSubgraph of G1, the_Vertices_of G1, the_Edges_of G1 by GLIB_000:34;

then G2 is inducedSubgraph of G1, the_Vertices_of G1, the_Edges_of G1 by A1, GLIB_006:16;

hence G2 is SimpleGraph of G1 by A2, Def9; :: thesis: verum

end;set E = the RepEdgeSelection of G1;

A2: the_Edges_of G1 = the RepEdgeSelection of G1 \ (G1 .loops()) by Th74;

G1 is inducedSubgraph of G1,(the_Vertices_of G1) by GLIB_006:15;

then G1 is inducedSubgraph of G1, the_Vertices_of G1, the_Edges_of G1 by GLIB_000:34;

then G2 is inducedSubgraph of G1, the_Vertices_of G1, the_Edges_of G1 by A1, GLIB_006:16;

hence G2 is SimpleGraph of G1 by A2, Def9; :: thesis: verum

then consider E being RepEdgeSelection of G1 such that

A3: G2 is inducedSubgraph of G1, the_Vertices_of G1,E \ (G1 .loops()) by Def9;

the_Edges_of G1 = E \ (G1 .loops()) by Th74;

then G2 is inducedSubgraph of G1,(the_Vertices_of G1) by A3, GLIB_000:34;

hence G1 == G2 by GLIB_000:94; :: thesis: verum