
theorem Th89:
for G being finite SimpleGraph holds order Mycielskian G = 2*(order G) + 1
proof
 let G be finite SimpleGraph;
 set uG = union G; set MG = Mycielskian G;
A1: card [:uG,{uG}:] = order G by CARD_1:69;
A2: uG misses [:uG,{uG}:] proof
    assume uG meets [:uG,{uG}:];
    then consider a being object such that
   A3: a in uG and
   A4: a in [:uG,{uG}:] by XBOOLE_0:3;
      consider x,y being object such that x in uG and
   A5: y in {uG} and
   A6: a = [x,y] by A4,ZFMISC_1:def 2;
      y = uG by A5,TARSKI:def 1;
     hence contradiction by A6,A3,Th1;
   end;
A7: now assume uG in (uG) \/ [:uG,{uG}:];
     then uG in uG or uG in [:uG,{uG}:] by XBOOLE_0:def 3;
     then consider x,y being object such that x in uG and
   A8: y in {uG} and
   A9: uG = [x,y] by ZFMISC_1:def 2;
     y = uG by A8,TARSKI:def 1;
    hence contradiction by A9,Th2;
   end;
 thus order MG = card ((uG) \/ [:uG,{uG}:] \/ {uG}) by Th86
  .= card ((uG) \/ [:uG,{uG}:]) + 1 by A7,CARD_2:41
  .= (card uG) + (card [:uG,{uG}:]) + 1 by A2,CARD_2:40
  .= 2*(order G) + 1 by A1;
end;
