
theorem Th57:
for G being with_finite_clique# SimpleGraph
  st Edges G <> {} holds clique# G >= 2
proof
  let G be with_finite_clique# SimpleGraph such that
A1: Edges G <> {};
  consider e being object such that
A2: e in Edges G by A1,XBOOLE_0:def 1;
  consider x, y being set such that
A3: x <> y and
A4: x in Vertices G and
A5: y in Vertices G and
A6: e = {x, y} by A2,Th11;
   reconsider S = G SubgraphInducedBy {x,y} as finite Clique of G by A6,A2,Th56
;
A7: Vertices S = (Vertices G) /\ {x,y} by Th45;
A8: {x,y} c= Vertices G by A4,A5,ZFMISC_1:32;
   Vertices S = {x,y} by A7,A8,XBOOLE_1:28;
   then order S = 2 by A3,CARD_2:57;
  hence clique# G >= 2 by Def15;
end;
