let G1, G2 be _Graph; :: thesis: for G3 being SimpleGraph of G1 st G2 == G3 holds

G2 is SimpleGraph of G1

let G3 be SimpleGraph of G1; :: thesis: ( G2 == G3 implies G2 is SimpleGraph of G1 )

consider E being RepEdgeSelection of G1 such that

A1: G3 is inducedSubgraph of G1, the_Vertices_of G1,E \ (G1 .loops()) by Def9;

assume G2 == G3 ; :: thesis: G2 is SimpleGraph of G1

then G2 is inducedSubgraph of G1, the_Vertices_of G1,E \ (G1 .loops()) by A1, GLIB_006:16;

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

G2 is SimpleGraph of G1

let G3 be SimpleGraph of G1; :: thesis: ( G2 == G3 implies G2 is SimpleGraph of G1 )

consider E being RepEdgeSelection of G1 such that

A1: G3 is inducedSubgraph of G1, the_Vertices_of G1,E \ (G1 .loops()) by Def9;

assume G2 == G3 ; :: thesis: G2 is SimpleGraph of G1

then G2 is inducedSubgraph of G1, the_Vertices_of G1,E \ (G1 .loops()) by A1, GLIB_006:16;

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