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

A2: now :: thesis: for x being object st x in {0} holds
(0 .--> 1) . x in {0,1}

let x be object ; :: 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_1:72;
hence (0 .--> 1) . x in {0,1} by TARSKI:def 2; :: thesis: verum

end;

A3: now :: thesis: for x being object st x in {0} holds
(0 .--> 0) . x in {0,1}

let x be object ; :: 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_1:72;
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_2:3;
dom (0 .--> 0) = {0} ;
then reconsider S = 0 .--> 0 as Function of {0},{0,1} by A3, FUNCT_2:3;
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 )
A4: (the_Source_of (createGraph ({0,1},{0},S,T))) . 0 = 0 by FUNCOP_1:72;

now :: thesis: for e1, e2, v1, v2 being object st e1 Joins v1,v2, createGraph ({0,1},{0},S,T) & e2 Joins v1,v2, createGraph ({0,1},{0},S,T) holds
not e1 <> e2

let e1, e2, v1, v2 be object ; :: 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 A6;
then A8: e1 = 0 by TARSKI:def 1;
assume A9: e1 <> e2 ; :: thesis: contradiction
e2 in {0} by A7;
hence contradiction by A9, A8, TARSKI:def 1; :: thesis: verum

end;

then A10: createGraph ({0,1},{0},S,T) is non-multi ;

now :: thesis: not card (the_Vertices_of (createGraph ({0,1},{0},S,T))) = 1

assume card (the_Vertices_of (createGraph ({0,1},{0},S,T))) = 1 ; :: thesis: contradiction
then ex x being object st the_Vertices_of (createGraph ({0,1},{0},S,T)) = {x} by CARD_2:42;
hence contradiction by ZFMISC_1:5; :: thesis: verum

end;

hence ( not createGraph ({0,1},{0},S,T) is _trivial & createGraph ({0,1},{0},S,T) is _finite ) ; :: thesis: ( createGraph ({0,1},{0},S,T) is simple & createGraph ({0,1},{0},S,T) is complete )
A12: (the_Target_of (createGraph ({0,1},{0},S,T))) . 0 = 1 by FUNCOP_1:72;
0 in the_Edges_of (createGraph ({0,1},{0},S,T)) by TARSKI:def 1;
then A13: 0 Joins 0 ,1, createGraph ({0,1},{0},S,T) by A4, A12;

now :: thesis: for v, e being object holds not e Joins v,v, createGraph ({0,1},{0},S,T)

let v, e be object ; :: 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_000:13;
e in the_Edges_of (createGraph ({0,1},{0},S,T)) by A14;
then e Joins 0 ,1, createGraph ({0,1},{0},S,T) by A13, TARSKI:def 1;
then ( ( 0 = v & 1 = v ) or ( 0 = v & 1 = v ) ) by A14;
hence contradiction ; :: thesis: verum

end;

then createGraph ({0,1},{0},S,T) is loopless by GLIB_000:18;
hence createGraph ({0,1},{0},S,T) is simple by A10; :: thesis: createGraph ({0,1},{0},S,T) is complete

now :: thesis: for u, v being Vertex of (createGraph ({0,1},{0},S,T)) st u <> v holds
u,v are_adjacent

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 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 TARSKI:def 1;
v = 1 by A15, A16, TARSKI:def 2;
then 0 Joins u,v, createGraph ({0,1},{0},S,T) by A4, A12, A16, A17;
hence u,v are_adjacent ; :: 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 TARSKI:def 1;
v = 0 by A15, A18, TARSKI:def 2;
then 0 Joins v,u, createGraph ({0,1},{0},S,T) by A4, A12, A18, A19;
hence u,v are_adjacent by Def3; :: thesis: verum

end;

end;

end;

hence createGraph ({0,1},{0},S,T) is complete ; :: thesis: verum