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

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

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

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

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

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

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

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

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

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;

hence G2 is removeLoops of G1 by A1, GLIB_006:16; :: thesis: verum

end;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;

hence G2 is removeLoops of G1 by A1, GLIB_006:16; :: thesis: verum

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

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