
theorem Th51:
for G being SimpleGraph, x, y being set
 st x in Vertices G & y in Vertices G & {x,y} in G
  holds { {}, {x}, {y}, {x,y} } is Clique of G
proof
 let G be SimpleGraph, x, y be set such that
A1: x in Vertices G and
A2: y in Vertices G and
A3: {x,y} in G;
  set C = CompleteSGraph {x,y};
A4: C = { {}, {x}, {y}, {x,y} } by Th37;
   C c= G proof
     let a be object;
     assume A5: a in C;
     per cases by A5,A4,ENUMSET1:def 2;
     suppose a = {};
      hence thesis by Th20;
     end;
     suppose a = {x};
       hence thesis by A1,Th24;
     end;
     suppose a = {y};
       hence thesis by A2,Th24;
     end;
     suppose a = {x,y};
       hence thesis by A3;
     end;
   end;
 hence { {}, {x}, {y}, {x,y} } is Clique of G by Th37;
end;
