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

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

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

then consider E being RepEdgeSelection of G1 such that

A2: G2 is inducedSubgraph of G1, the_Vertices_of G1,E by Def7;

E = E \ (G1 .loops()) ;

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

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

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

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

assume
G2 is removeParallelEdges of G1
; :: thesis: G2 is SimpleGraph of G1assume
G2 is SimpleGraph of G1
; :: thesis: G2 is removeParallelEdges of G1

then consider E being RepEdgeSelection of G1 such that

A1: G2 is inducedSubgraph of G1, the_Vertices_of G1,E \ (G1 .loops()) by Def9;

thus G2 is removeParallelEdges of G1 by A1, Def7; :: thesis: verum

end;then consider E being RepEdgeSelection of G1 such that

A1: G2 is inducedSubgraph of G1, the_Vertices_of G1,E \ (G1 .loops()) by Def9;

thus G2 is removeParallelEdges of G1 by A1, Def7; :: thesis: verum

then consider E being RepEdgeSelection of G1 such that

A2: G2 is inducedSubgraph of G1, the_Vertices_of G1,E by Def7;

E = E \ (G1 .loops()) ;

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