set V = {0 ,1};
set E = {0 };
set S = 0 .--> 0 ;
set T = 0 .--> 1;
A5: ( dom (0 .--> 0 ) = {0 } & dom (0 .--> 1) = {0 } ) by FUNCOP_1:19;

now

let x be set ; :: thesis: ( x in {0 } implies (0 .--> 0 ) . x in {0 ,1} )
assume x in {0 } ; :: thesis: (0 .--> 0 ) . x in {0 ,1}
then x = 0 by TARSKI:def 1;
then (0 .--> 0 ) . x = 0 by FUNCOP_1:87;
hence (0 .--> 0 ) . x in {0 ,1} by TARSKI:def 2; :: thesis: verum

end;

then reconsider S = 0 .--> 0 as Function of {0 },{0 ,1} by A5, FUNCT_2:5;

now

let x be set ; :: thesis: ( x in {0 } implies (0 .--> 1) . x in {0 ,1} )
assume x in {0 } ; :: thesis: (0 .--> 1) . x in {0 ,1}
then x = 0 by TARSKI:def 1;
then (0 .--> 1) . x = 1 by FUNCOP_1:87;
hence (0 .--> 1) . x in {0 ,1} by TARSKI:def 2; :: thesis: verum

end;

then reconsider T = 0 .--> 1 as Function of {0 },{0 ,1} by A5, FUNCT_2:5;
set G = createGraph {0 ,1},{0 },S,T;
A6: ( the_Vertices_of (createGraph {0 ,1},{0 },S,T) = {0 ,1} & the_Edges_of (createGraph {0 ,1},{0 },S,T) = {0 } & the_Source_of (createGraph {0 ,1},{0 },S,T) = S & the_Target_of (createGraph {0 ,1},{0 },S,T) = T ) by GLIB_000:8;
take createGraph {0 ,1},{0 },S,T ; :: thesis: ( not createGraph {0 ,1},{0 },S,T is trivial & createGraph {0 ,1},{0 },S,T is finite & createGraph {0 ,1},{0 },S,T is simple & createGraph {0 ,1},{0 },S,T is chordal )

now

assume card (the_Vertices_of (createGraph {0 ,1},{0 },S,T)) = 1 ; :: thesis: contradiction
then consider x being set such that
A7: the_Vertices_of (createGraph {0 ,1},{0 },S,T) = {x} by CARD_2:60;
thus contradiction by A6, A7, ZFMISC_1:9; :: thesis: verum

end;

hence ( not createGraph {0 ,1},{0 },S,T is trivial & createGraph {0 ,1},{0 },S,T is finite ) by GLIB_000:def 21; :: thesis: ( createGraph {0 ,1},{0 },S,T is simple & createGraph {0 ,1},{0 },S,T is chordal )
A8: (the_Source_of (createGraph {0 ,1},{0 },S,T)) . 0 = 0 by A6, FUNCOP_1:87;
A9: (the_Target_of (createGraph {0 ,1},{0 },S,T)) . 0 = 1 by A6, FUNCOP_1:87;
0 in the_Edges_of (createGraph {0 ,1},{0 },S,T) by A6, TARSKI:def 1;
then A10: 0 Joins 0 ,1, createGraph {0 ,1},{0 },S,T by A8, A9, GLIB_000:def 15;

now

let v, e be set ; :: thesis: not e Joins v,v, createGraph {0 ,1},{0 },S,T
assume A11: e Joins v,v, createGraph {0 ,1},{0 },S,T ; :: thesis: contradiction
reconsider v = v as Vertex of (createGraph {0 ,1},{0 },S,T) by A11, GLIB_000:16;
e in the_Edges_of (createGraph {0 ,1},{0 },S,T) by A11, GLIB_000:def 15;
then e Joins 0 ,1, createGraph {0 ,1},{0 },S,T by A6, A10, TARSKI:def 1;
then ( ( 0 = v & 1 = v ) or ( 0 = v & 1 = v ) ) by A11, GLIB_000:18;
hence contradiction ; :: thesis: verum

end;

then A12: createGraph {0 ,1},{0 },S,T is loopless by GLIB_000:21;

now

let e1, e2, v1, v2 be set ; :: thesis: ( e1 Joins v1,v2, createGraph {0 ,1},{0 },S,T & e2 Joins v1,v2, createGraph {0 ,1},{0 },S,T implies not e1 <> e2 )
assume A13: ( e1 Joins v1,v2, createGraph {0 ,1},{0 },S,T & e2 Joins v1,v2, createGraph {0 ,1},{0 },S,T ) ; :: thesis: not e1 <> e2
assume A14: e1 <> e2 ; :: thesis: contradiction
( e1 in {0 } & e2 in {0 } ) by A6, A13, GLIB_000:def 15;
then ( e1 = 0 & e2 = 0 ) by TARSKI:def 1;
hence contradiction by A14; :: thesis: verum

end;

then createGraph {0 ,1},{0 },S,T is non-multi by GLIB_000:def 22;
hence createGraph {0 ,1},{0 },S,T is simple by A12; :: thesis: createGraph {0 ,1},{0 },S,T is chordal
card (the_Vertices_of (createGraph {0 ,1},{0 },S,T)) = 2 by A6, CARD_2:76;
hence createGraph {0 ,1},{0 },S,T is chordal by Th96; :: thesis: verum