let G1 be _Graph; :: thesis: for G2 being removeLoops of G1
for G3 being DSimpleGraph of G1 holds G3 is removeDParallelEdges of G2
let G2 be removeLoops of G1; :: thesis: for G3 being DSimpleGraph of G1 holds G3 is removeDParallelEdges of G2
let G3 be DSimpleGraph of G1; :: thesis: G3 is removeDParallelEdges of G2
consider E being RepDEdgeSelection of G1 such that
A1: G3 is inducedSubgraph of G1, the_Vertices_of G1,E \ (G1 .loops()) by Def10;
A2: ( the_Vertices_of G1 c= the_Vertices_of G1 & the_Edges_of G1 = G1 .edgesBetween (the_Vertices_of G1) ) by GLIB_000:34;
then A3: ( the_Vertices_of G3 = the_Vertices_of G1 & the_Edges_of G3 = E \ (G1 .loops()) ) by A1, GLIB_000:def 37;
A4: ( the_Vertices_of G2 = the_Vertices_of G1 & the_Edges_of G2 = (the_Edges_of G1) \ (G1 .loops()) ) by GLIB_000:53;
set E2 = E \ (G1 .loops());
A5: E \ (G1 .loops()) c= the_Edges_of G2 by A4, XBOOLE_1:33;
now :: thesis: for v, w, e0 being object st e0 DJoins v,w,G2 holds
ex e being object st
( e DJoins v,w,G2 & e in E \ (G1 .loops()) & ( for e9 being object st e9 DJoins v,w,G2 & e9 in E \ (G1 .loops()) holds
e9 = e ) )
let v, w, e0 be object ; :: thesis: ( e0 DJoins v,w,G2 implies ex e being object st
( e DJoins v,w,G2 & e in E \ (G1 .loops()) & ( for e9 being object st e9 DJoins v,w,G2 & e9 in E \ (G1 .loops()) holds
e9 = e ) ) )
A6: ( v is set & w is set ) by TARSKI:1;
assume A7: e0 DJoins v,w,G2 ; :: thesis: ex e being object st
( e DJoins v,w,G2 & e in E \ (G1 .loops()) & ( for e9 being object st e9 DJoins v,w,G2 & e9 in E \ (G1 .loops()) holds
e9 = e ) )
then A8: e0 DJoins v,w,G1 by A6, GLIB_000:72;
then consider e being object such that
A9: ( e DJoins v,w,G1 & e in E ) and
A10: for e9 being object st e9 DJoins v,w,G1 & e9 in E holds
e9 = e by Def6;
take e = e; :: thesis: ( e DJoins v,w,G2 & e in E \ (G1 .loops()) & ( for e9 being object st e9 DJoins 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 DJoins u,u,G1 by Th45;
( v = u & w = u ) by A9, A12, Th6;
then A13: e0 in G1 .loops() by A8, Th45;
e0 in the_Edges_of G2 by A7, GLIB_000:def 14;
hence contradiction by A4, A13, XBOOLE_0:def 5; :: thesis: verum
end;
then e in the_Edges_of G2 by A4, A9, XBOOLE_0:def 5;
hence e DJoins v,w,G2 by A6, A9, GLIB_000:73; :: thesis: ( e in E \ (G1 .loops()) & ( for e9 being object st e9 DJoins v,w,G2 & e9 in E \ (G1 .loops()) holds
e9 = e ) )
thus e in E \ (G1 .loops()) by A9, A11, XBOOLE_0:def 5; :: thesis: for e9 being object st e9 DJoins v,w,G2 & e9 in E \ (G1 .loops()) holds
e9 = e
let e9 be object ; :: thesis: ( e9 DJoins v,w,G2 & e9 in E \ (G1 .loops()) implies e9 = e )
assume ( e9 DJoins v,w,G2 & e9 in E \ (G1 .loops()) ) ; :: thesis: e9 = e
then ( e9 DJoins v,w,G1 & e9 in E \ (G1 .loops()) ) by A6, GLIB_000:72;
then ( e9 DJoins v,w,G1 & e9 in E ) by XBOOLE_1:36, TARSKI:def 3;
hence e9 = e by A10; :: thesis: verum
end;
then reconsider E2 = E \ (G1 .loops()) as RepDEdgeSelection of G2 by A5, Def6;
A14: the_Vertices_of G3 = the_Vertices_of G2 by A3, A4;
E \ (G1 .loops()) c= (the_Edges_of G1) \ (G1 .loops()) by XBOOLE_1:33;
then A15: G3 is Subgraph of G2 by A1, A2, GLIB_000:46;
( the_Vertices_of G2 c= the_Vertices_of G2 & the_Edges_of G2 = G2 .edgesBetween (the_Vertices_of G2) ) by GLIB_000:34;
then G3 is inducedSubgraph of G2, the_Vertices_of G2,E2 by A3, A14, A15, GLIB_000:def 37;
hence G3 is removeDParallelEdges of G2 by Def8; :: thesis: verum