set V = {1,2};
set E = {} ;
reconsider S = {} as Function of {},{1,2} by RELSET_1:12;
set G = createGraph ({1,2},{},S,S);
take createGraph ({1,2},{},S,S) ; :: thesis: ( createGraph ({1,2},{},S,S) is _finite & not createGraph ({1,2},{},S,S) is _trivial & createGraph ({1,2},{},S,S) is simple )
A2: the_Edges_of (createGraph ({1,2},{},S,S)) = {} by FINSEQ_4:76;
A3: the_Vertices_of (createGraph ({1,2},{},S,S)) = {1,2} by FINSEQ_4:76;
hence createGraph ({1,2},{},S,S) is _finite by A2; :: thesis: ( not createGraph ({1,2},{},S,S) is _trivial & createGraph ({1,2},{},S,S) is simple )
card (the_Vertices_of (createGraph ({1,2},{},S,S))) <> 1 by A3, CARD_2:57;
hence not createGraph ({1,2},{},S,S) is _trivial ; :: thesis: createGraph ({1,2},{},S,S) is simple
for e being object holds
( not e in the_Edges_of (createGraph ({1,2},{},S,S)) or not (the_Source_of (createGraph ({1,2},{},S,S))) . e = (the_Target_of (createGraph ({1,2},{},S,S))) . e ) by FINSEQ_4:76;
then A4: createGraph ({1,2},{},S,S) is loopless ;
for e1, e2, v1, v2 being object st e1 Joins v1,v2, createGraph ({1,2},{},S,S) & e2 Joins v1,v2, createGraph ({1,2},{},S,S) holds
e1 = e2 by A2;
then createGraph ({1,2},{},S,S) is non-multi ;
hence createGraph ({1,2},{},S,S) is simple by A4; :: thesis: verum