let G1 be _Graph; :: thesis:
let G2 be removeParallelEdges of G1; :: thesis:
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 (the_Vertices_of G1) ) by GLIB_000:34;
then A2: the_Edges_of G2 = E by A1, GLIB_000:def 37;

hereby :: thesis:

assume A3: G1 is chordal ; :: thesis:

now :: thesis:

let P2 be Walk of G2; :: thesis:
assume A4: ( P2 .length() > 3 & P2 is Cycle-like ) ; :: thesis:

now :: thesis:

reconsider P1 = P2 as Walk of G1 by GLIB_001:167;
( P1 .length() > 3 & P1 is Cycle-like ) by A4, GLIB_001:114, GLIB_006:24;
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 A3, CHORD:def 11, CHORD:def 10;
take m = m; :: thesis:
take n = n; :: thesis:
thus ( m + 2 < n & n <= len P2 & P2 . m <> P2 . n ) by A5; :: thesis:

hereby :: thesis:

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 A8, Def5;
take e = e; :: thesis:
thus e Joins P2 . m,P2 . n,G2 by A2, A9, GLIB_000:73; :: thesis:

end;

let f be object ; :: thesis:
assume f in P2 .edges() ; :: thesis:
then f in P1 .edges() by GLIB_001:110;
hence not f Joins P2 . m,P2 . n,G2 by A7, GLIB_000:72; :: thesis:

end;

hence P2 is chordal by CHORD:def 10; :: thesis:

end;

hence G2 is chordal by CHORD:def 11; :: thesis:

end;

assume A10: G2 is chordal ; :: thesis:

now :: thesis:

let P1 be Walk of G1; :: thesis:
assume A11: ( P1 .length() > 3 & P1 is Cycle-like ) ; :: thesis:
A12: len P1 <> 5

proof

assume len P1 = 5 ; :: thesis:
then (2 * (P1 .length())) + 1 = (2 * 2) + 1 by GLIB_001:112;
hence contradiction by A11; :: thesis:

end;

now :: thesis:

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 A11, A12, A13, Th30, Th32;
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 A10, A11, CHORD:def 11, CHORD:def 10;
take m = m; :: thesis:
take n = n; :: thesis:
A18: len P1 = (2 * (P1 .length())) + 1 by GLIB_001:112
.= len P2 by A14, GLIB_001:112 ;
hence ( m + 2 < n & n <= len P1 ) by A15; :: thesis:
A19: ( P1 . n = P2 . n & P1 . m = P2 . m ) by A13, Th29;
hence P1 . m <> P1 . n by A15; :: thesis:
thus ex e being object st e Joins P1 . m,P1 . n,G1 by A16, A19, GLIB_000:72; :: thesis:
let f be object ; :: thesis:
assume f in P1 .edges() ; :: thesis:
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 A20, GLIB_001:def 3;
assume A22: f Joins P1 . m,P1 . n,G1 ; :: thesis:
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 A2, A19, A23, GLIB_000:73;
A26: k < len P2 by A18, A20;
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 A13, Th29;
then A28: P2 . (k + 1) Joins P1 . k,P1 . (k + 2),G1 by A27, GLIB_000:72;
A29: P2 . (k + 1) in E by A2, A27, GLIB_000:def 13;
( ( P1 . m = P1 . k & P1 . n = P1 . (k + 2) ) or ( P1 . m = P1 . (k + 2) & P1 . n = P1 . k ) ) by A21, A22, GLIB_000:15;
then P2 . (k + 1) = f2 by A24, A28, A29, GLIB_000:14;
hence contradiction by A17, A25, A26, GLIB_001:100; :: thesis:

end;

hence P1 is chordal by CHORD:def 10; :: thesis:

end;

hence G1 is chordal by CHORD:def 11; :: thesis: