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