set V = {0 ,1};
set E = {0 };
set S = 0 .--> 0 ;
set T = 0 .--> 1;
A1: dom (0 .--> 1) = {0 } by FUNCOP_1:19;

A2: 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;

A3: 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;

reconsider T = 0 .--> 1 as Function of {0 },{0 ,1} by A1, A2, FUNCT_2:5;
dom (0 .--> 0 ) = {0 } by FUNCOP_1:19;
then reconsider S = 0 .--> 0 as Function of {0 },{0 ,1} by A3, FUNCT_2:5;
set G = createGraph {0 ,1},{0 },S,T;
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 complete )
the_Source_of (createGraph {0 ,1},{0 },S,T) = S by GLIB_000:8;
then A4: (the_Source_of (createGraph {0 ,1},{0 },S,T)) . 0 = 0 by FUNCOP_1:87;
A5: the_Edges_of (createGraph {0 ,1},{0 },S,T) = {0 } by GLIB_000:8;

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 that
A6: e1 Joins v1,v2, createGraph {0 ,1},{0 },S,T and
A7: e2 Joins v1,v2, createGraph {0 ,1},{0 },S,T ; :: thesis: not e1 <> e2
e1 in {0 } by A5, A6, GLIB_000:def 15;
then A8: e1 = 0 by TARSKI:def 1;
assume A9: e1 <> e2 ; :: thesis: contradiction
e2 in {0 } by A5, A7, GLIB_000:def 15;
hence contradiction by A9, A8, TARSKI:def 1; :: thesis: verum

end;

then A10: createGraph {0 ,1},{0 },S,T is non-multi by GLIB_000:def 22;
A11: the_Vertices_of (createGraph {0 ,1},{0 },S,T) = {0 ,1} by GLIB_000:8;

now

assume card (the_Vertices_of (createGraph {0 ,1},{0 },S,T)) = 1 ; :: thesis: contradiction
then ex x being set st the_Vertices_of (createGraph {0 ,1},{0 },S,T) = {x} by CARD_2:60;
hence contradiction by A11, 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 complete )
the_Target_of (createGraph {0 ,1},{0 },S,T) = T by GLIB_000:8;
then A12: (the_Target_of (createGraph {0 ,1},{0 },S,T)) . 0 = 1 by FUNCOP_1:87;
0 in the_Edges_of (createGraph {0 ,1},{0 },S,T) by A5, TARSKI:def 1;
then A13: 0 Joins 0 ,1, createGraph {0 ,1},{0 },S,T by A4, A12, GLIB_000:def 15;

now

let v, e be set ; :: thesis: not e Joins v,v, createGraph {0 ,1},{0 },S,T
assume A14: 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 A14, GLIB_000:16;
e in the_Edges_of (createGraph {0 ,1},{0 },S,T) by A14, GLIB_000:def 15;
then e Joins 0 ,1, createGraph {0 ,1},{0 },S,T by A5, A13, TARSKI:def 1;
then ( ( 0 = v & 1 = v ) or ( 0 = v & 1 = v ) ) by A14, GLIB_000:18;
hence contradiction ; :: thesis: verum

end;

then createGraph {0 ,1},{0 },S,T is loopless by GLIB_000:21;
hence createGraph {0 ,1},{0 },S,T is simple by A10; :: thesis: createGraph {0 ,1},{0 },S,T is complete

now

let u, v be Vertex of (createGraph {0 ,1},{0 },S,T); :: thesis: ( u <> v implies b₁,b₂ are_adjacent )
assume A15: u <> v ; :: thesis: b₁,b₂ are_adjacent

per cases ( u = 0 or u = 1 ) by A11, TARSKI:def 2;

suppose A16: u = 0 ; :: thesis: b₁,b₂ are_adjacent

A17: 0 in the_Edges_of (createGraph {0 ,1},{0 },S,T) by A5, TARSKI:def 1;
v = 1 by A11, A15, A16, TARSKI:def 2;
then 0 Joins u,v, createGraph {0 ,1},{0 },S,T by A4, A12, A16, A17, GLIB_000:def 15;
hence u,v are_adjacent by Def3; :: thesis: verum

end;

suppose A18: u = 1 ; :: thesis: b₁,b₂ are_adjacent

A19: 0 in the_Edges_of (createGraph {0 ,1},{0 },S,T) by A5, TARSKI:def 1;
v = 0 by A11, A15, A18, TARSKI:def 2;
then 0 Joins v,u, createGraph {0 ,1},{0 },S,T by A4, A12, A18, A19, GLIB_000:def 15;
hence u,v are_adjacent by Def3; :: thesis: verum

end;

end;

end;

hence createGraph {0 ,1},{0 },S,T is complete by Def6; :: thesis: verum