
theorem Th32:
for G being SimpleGraph
  st (for x, y being set st x in Vertices G & y in Vertices G
        holds {x, y} in G)
   holds G = CompleteSGraph Vertices G
proof
  let G be SimpleGraph such that
A1: for x, y being set st x in Vertices G & y in Vertices G
        holds {x, y} in G;
   set C = { V where V is finite Subset of Vertices G : card V <= 2};
   C = G proof
    thus C c= G proof
      let a be object;
      assume a in C;
       then consider V being finite Subset of Vertices G such that
      A2: a = V and
      A3: card V <= 2;
       card V = 0 or ... or card V = 2 by A3;
       then per cases;
       suppose card V = 0;
         then V = {};
        hence a in G by A2,Th20;
       end;
       suppose card V = 1;
         then consider c being object such that
       A4: V = {c} by CARD_2:42;
         c in V by A4,TARSKI:def 1;
         then {c,c} in G by A1;
        hence a in G by A4,A2,ENUMSET1:29;
       end;
       suppose card V = 2;
         then consider c,d being object such that
          c <> d and
       A5: V = {c, d} by CARD_2:60;
         c in V & d in V by A5,TARSKI:def 2;
        hence a in G by A1,A5,A2;
       end;
    end;
    thus G c= C proof
     let a be object;
     assume A6: a in G;
       then reconsider aa = a as finite set;
     A7: card aa <= 1+1 by A6,Th21;
       aa c= union G by A6,ZFMISC_1:74;
     hence a in C by A7;
    end;
   end;
  hence G = CompleteSGraph Vertices G;
end;
