let G1 be _Graph; :: thesis: for G2 being removeLoops of G1
for G3 being SimpleGraph of G1 holds G3 is removeParallelEdges of G2

let G2 be removeLoops of G1; :: thesis: for G3 being SimpleGraph of G1 holds G3 is removeParallelEdges of G2
let G3 be SimpleGraph of G1; :: thesis: G3 is removeParallelEdges of G2
consider E being RepEdgeSelection of G1 such that
A1: G3 is inducedSubgraph of G1, the_Vertices_of G1,E \ (G1 .loops()) by Def9;
A2: ( the_Vertices_of G1 c= the_Vertices_of G1 & the_Edges_of G1 = G1 .edgesBetween () ) by GLIB_000:34;
then A3: ( the_Vertices_of G3 = the_Vertices_of G1 & the_Edges_of G3 = E \ (G1 .loops()) ) by ;
A4: ( the_Vertices_of G2 = the_Vertices_of G1 & the_Edges_of G2 = () \ (G1 .loops()) ) by GLIB_000:53;
set E2 = E \ (G1 .loops());
A5: E \ (G1 .loops()) c= the_Edges_of G2 by ;
now :: thesis: for v, w, e0 being object st e0 Joins v,w,G2 holds
ex e being object st
( e Joins v,w,G2 & e in E \ (G1 .loops()) & ( for e9 being object st e9 Joins v,w,G2 & e9 in E \ (G1 .loops()) holds
e9 = e ) )
let v, w, e0 be object ; :: thesis: ( e0 Joins v,w,G2 implies ex e being object st
( e Joins v,w,G2 & e in E \ (G1 .loops()) & ( for e9 being object st e9 Joins v,w,G2 & e9 in E \ (G1 .loops()) holds
e9 = e ) ) )

A6: ( v is set & w is set ) by TARSKI:1;
assume A7: e0 Joins v,w,G2 ; :: thesis: ex e being object st
( e Joins v,w,G2 & e in E \ (G1 .loops()) & ( for e9 being object st e9 Joins v,w,G2 & e9 in E \ (G1 .loops()) holds
e9 = e ) )

then A8: e0 Joins v,w,G1 by ;
then consider e being object such that
A9: ( e Joins v,w,G1 & e in E ) and
A10: for e9 being object st e9 Joins v,w,G1 & e9 in E holds
e9 = e by Def5;
take e = e; :: thesis: ( e Joins v,w,G2 & e in E \ (G1 .loops()) & ( for e9 being object st e9 Joins v,w,G2 & e9 in E \ (G1 .loops()) holds
e9 = e ) )

A11: not e in G1 .loops()
proof
assume e in G1 .loops() ; :: thesis: contradiction
then consider u being object such that
A12: e Joins u,u,G1 by Def2;
( v = u & w = u ) by ;
then A13: e0 in G1 .loops() by ;
e0 in the_Edges_of G2 by ;
hence contradiction by A4, A13, XBOOLE_0:def 5; :: thesis: verum
end;
e in the_Edges_of G2 by ;
hence e Joins v,w,G2 by ; :: thesis: ( e in E \ (G1 .loops()) & ( for e9 being object st e9 Joins v,w,G2 & e9 in E \ (G1 .loops()) holds
e9 = e ) )

thus e in E \ (G1 .loops()) by ; :: thesis: for e9 being object st e9 Joins v,w,G2 & e9 in E \ (G1 .loops()) holds
e9 = e

let e9 be object ; :: thesis: ( e9 Joins v,w,G2 & e9 in E \ (G1 .loops()) implies e9 = e )
assume ( e9 Joins v,w,G2 & e9 in E \ (G1 .loops()) ) ; :: thesis: e9 = e
then ( e9 Joins v,w,G1 & e9 in E \ (G1 .loops()) ) by ;
then ( e9 Joins v,w,G1 & e9 in E ) by ;
hence e9 = e by A10; :: thesis: verum
end;
then reconsider E2 = E \ (G1 .loops()) as RepEdgeSelection of G2 by ;
A14: the_Vertices_of G3 = the_Vertices_of G2 by A3, A4;
E \ (G1 .loops()) c= () \ (G1 .loops()) by XBOOLE_1:33;
then A15: G3 is Subgraph of G2 by ;
( the_Vertices_of G2 c= the_Vertices_of G2 & the_Edges_of G2 = G2 .edgesBetween () ) by GLIB_000:34;
then G3 is inducedSubgraph of G2, the_Vertices_of G2,E2 by ;
hence G3 is removeParallelEdges of G2 by Def7; :: thesis: verum