let G1 be _Graph; :: thesis: for G2 being removeParallelEdges of G1 holds
( G1 is chordal iff G2 is chordal )

let G2 be removeParallelEdges of G1; :: thesis: ( G1 is chordal iff G2 is chordal )
consider E being RepEdgeSelection of G1 such that
A1: G2 is inducedSubgraph of G1, the_Vertices_of G1,E by Def7;
( the_Vertices_of G1 c= the_Vertices_of G1 & the_Edges_of G1 = G1 .edgesBetween () ) by GLIB_000:34;
then A2: the_Edges_of G2 = E by ;
hereby :: thesis: ( G2 is chordal implies G1 is chordal )
assume A3: G1 is chordal ; :: thesis: G2 is chordal
now :: thesis: for P2 being Walk of G2 st P2 .length() > 3 & P2 is Cycle-like holds
P2 is chordal
let P2 be Walk of G2; :: thesis: ( P2 .length() > 3 & P2 is Cycle-like implies P2 is chordal )
assume A4: ( P2 .length() > 3 & P2 is Cycle-like ) ; :: thesis: P2 is chordal
now :: thesis: ex m, n being odd Nat st
( m + 2 < n & n <= len P2 & P2 . m <> P2 . n & ex e being object st e Joins P2 . m,P2 . n,G2 & ( for f being object st f in P2 .edges() holds
not f Joins P2 . m,P2 . n,G2 ) )
reconsider P1 = P2 as Walk of G1 by GLIB_001:167;
( P1 .length() > 3 & P1 is Cycle-like ) by ;
then consider m, n being odd Nat such that
A5: ( m + 2 < n & n <= len P1 & P1 . m <> P1 . n ) and
A6: ex e being object st e Joins P1 . m,P1 . n,G1 and
A7: for f being object st f in P1 .edges() holds
not f Joins P1 . m,P1 . n,G1 by ;
take m = m; :: thesis: ex n being odd Nat st
( m + 2 < n & n <= len P2 & P2 . m <> P2 . n & ex e being object st e Joins P2 . m,P2 . n,G2 & ( for f being object st f in P2 .edges() holds
not f Joins P2 . m,P2 . n,G2 ) )

take n = n; :: thesis: ( m + 2 < n & n <= len P2 & P2 . m <> P2 . n & ex e being object st e Joins P2 . m,P2 . n,G2 & ( for f being object st f in P2 .edges() holds
not f Joins P2 . m,P2 . n,G2 ) )

thus ( m + 2 < n & n <= len P2 & P2 . m <> P2 . n ) by A5; :: thesis: ( ex e being object st e Joins P2 . m,P2 . n,G2 & ( for f being object st f in P2 .edges() holds
not f Joins P2 . m,P2 . n,G2 ) )

hereby :: thesis: for f being object st f in P2 .edges() holds
not f Joins P2 . m,P2 . n,G2
consider e0 being object such that
A8: e0 Joins P1 . m,P1 . n,G1 by A6;
consider e being object such that
A9: ( e Joins P1 . m,P1 . n,G1 & e in E ) and
for e9 being object st e9 Joins P1 . m,P1 . n,G1 & e9 in E holds
e9 = e by ;
take e = e; :: thesis: e Joins P2 . m,P2 . n,G2
thus e Joins P2 . m,P2 . n,G2 by ; :: thesis: verum
end;
let f be object ; :: thesis: ( f in P2 .edges() implies not f Joins P2 . m,P2 . n,G2 )
assume f in P2 .edges() ; :: thesis: not f Joins P2 . m,P2 . n,G2
then f in P1 .edges() by GLIB_001:110;
hence not f Joins P2 . m,P2 . n,G2 by ; :: thesis: verum
end;
hence P2 is chordal by CHORD:def 10; :: thesis: verum
end;
hence G2 is chordal by CHORD:def 11; :: thesis: verum
end;
assume A10: G2 is chordal ; :: thesis: G1 is chordal
now :: thesis: for P1 being Walk of G1 st P1 .length() > 3 & P1 is Cycle-like holds
P1 is chordal
let P1 be Walk of G1; :: thesis: ( P1 .length() > 3 & P1 is Cycle-like implies P1 is chordal )
assume A11: ( P1 .length() > 3 & P1 is Cycle-like ) ; :: thesis: P1 is chordal
A12: len P1 <> 5
proof
assume len P1 = 5 ; :: thesis: contradiction
then (2 * ()) + 1 = (2 * 2) + 1 by GLIB_001:112;
hence contradiction by A11; :: thesis: verum
end;
now :: thesis: ex m, n being odd Nat st
( m + 2 < n & n <= len P1 & P1 . m <> P1 . n & ex e being object st e Joins P1 . m,P1 . n,G1 & ( for f being object st f in P1 .edges() holds
not f Joins P1 . m,P1 . n,G1 ) )
consider P2 being Walk of G2 such that
A13: P1 .vertexSeq() = P2 .vertexSeq() by Th98;
A14: ( P1 .length() = P2 .length() & P2 is Cycle-like ) by ;
then consider m, n being odd Nat such that
A15: ( m + 2 < n & n <= len P2 & P2 . m <> P2 . n ) and
A16: ex e being object st e Joins P2 . m,P2 . n,G2 and
A17: for f being object st f in P2 .edges() holds
not f Joins P2 . m,P2 . n,G2 by ;
take m = m; :: thesis: ex n being odd Nat st
( m + 2 < n & n <= len P1 & P1 . m <> P1 . n & ex e being object st e Joins P1 . m,P1 . n,G1 & ( for f being object st f in P1 .edges() holds
not f Joins P1 . m,P1 . n,G1 ) )

take n = n; :: thesis: ( m + 2 < n & n <= len P1 & P1 . m <> P1 . n & ex e being object st e Joins P1 . m,P1 . n,G1 & ( for f being object st f in P1 .edges() holds
not f Joins P1 . m,P1 . n,G1 ) )

A18: len P1 = (2 * ()) + 1 by GLIB_001:112
.= len P2 by ;
hence ( m + 2 < n & n <= len P1 ) by A15; :: thesis: ( P1 . m <> P1 . n & ex e being object st e Joins P1 . m,P1 . n,G1 & ( for f being object st f in P1 .edges() holds
not f Joins P1 . m,P1 . n,G1 ) )

A19: ( P1 . n = P2 . n & P1 . m = P2 . m ) by ;
hence P1 . m <> P1 . n by A15; :: thesis: ( ex e being object st e Joins P1 . m,P1 . n,G1 & ( for f being object st f in P1 .edges() holds
not f Joins P1 . m,P1 . n,G1 ) )

thus ex e being object st e Joins P1 . m,P1 . n,G1 by ; :: thesis: for f being object st f in P1 .edges() holds
not f Joins P1 . m,P1 . n,G1

let f be object ; :: thesis: ( f in P1 .edges() implies not f Joins P1 . m,P1 . n,G1 )
assume f in P1 .edges() ; :: thesis: not f Joins P1 . m,P1 . n,G1
then consider k being odd Element of NAT such that
A20: ( k < len P1 & P1 . (k + 1) = f ) by GLIB_001:100;
A21: f Joins P1 . k,P1 . (k + 2),G1 by ;
assume A22: f Joins P1 . m,P1 . n,G1 ; :: thesis: contradiction
then consider f2 being object such that
A23: ( f2 Joins P1 . m,P1 . n,G1 & f2 in E ) and
A24: for f9 being object st f9 Joins P1 . m,P1 . n,G1 & f9 in E holds
f9 = f2 by Def5;
A25: f2 Joins P2 . m,P2 . n,G2 by ;
A26: k < len P2 by ;
then A27: P2 . (k + 1) Joins P2 . k,P2 . (k + 2),G2 by GLIB_001:def 3;
( P1 . k = P2 . k & P1 . (k + 2) = P2 . (k + 2) ) by ;
then A28: P2 . (k + 1) Joins P1 . k,P1 . (k + 2),G1 by ;
A29: P2 . (k + 1) in E by ;
( ( P1 . m = P1 . k & P1 . n = P1 . (k + 2) ) or ( P1 . m = P1 . (k + 2) & P1 . n = P1 . k ) ) by ;
then P2 . (k + 1) = f2 by ;
hence contradiction by A17, A25, A26, GLIB_001:100; :: thesis: verum
end;
hence P1 is chordal by CHORD:def 10; :: thesis: verum
end;
hence G1 is chordal by CHORD:def 11; :: thesis: verum