
theorem Th92:
for G being finite SimpleGraph holds size Mycielskian G = 3*(size G) + order G
proof
 let G be finite SimpleGraph;
 set uG = union G; set MG = Mycielskian G;
 set A = { {x,[y,uG]} where x, y is Element of uG : {x,y} in Edges G };
 set B = { {uG,[y,uG]} where y is Element of uG : y in union G };
 per cases;
 suppose A1: G is void;
  then A2: MG = {{},{uG}} by Th88;
  A3: size G = 0 by A1,Th17,CARD_1:27;
      size MG = 0 proof
       assume not thesis;
        then Edges MG <> {};
        then consider e being object such that
      A4: e in Edges MG by XBOOLE_0:def 1;
         consider x, y being set such that
      A5: x <> y and x in Vertices MG & y in Vertices MG and
      A6: e = {x, y} by A4,Th11;
          e = {} or e = {uG} by A2,A4,TARSKI:def 2;
        hence thesis by A6,A5,ZFMISC_1:5;
      end;
   hence thesis by A1,A3;
 end;
 suppose G is non void;
   then reconsider uGf = uG as non empty set by Th28;
A7: uGf is finite;
   deffunc FB(set) = {uG,[$1,uG]};
   { FB(x) where x is Element of uGf : x in uGf } is finite
      from FRAENKEL:sch 21(A7);
   then reconsider B as finite set;
A8: uG is finite;
   deffunc FA(set,set) = {$1,[$2,uG]};
   set AA = { FA(x,y) where x is Element of uGf, y is Element of uGf
     : x in uG & y in uG };
A9: AA is finite from FRAENKEL:sch 22(A8, A8);
   A c= AA proof
     let a be object;
     assume a in A;
     then consider x, y being Element of uG such that
   A10: a = {x,[y,uG]} and {x,y} in Edges G;
     thus a in AA by A10;
   end;
   then reconsider A as finite set by A9;
A11: card B = order G by Th10;
A12: card A = 2 * size G by Th15;
A13: now assume B meets (Edges G) \/ A;
     then consider a being object such that
   A14: a in B and
   A15: a in (Edges G) \/ A by XBOOLE_0:3;
       consider y being Element of uG such that
   A16: a = {uG,[y,uG]} and y in union G by A14;
       per cases by A15,XBOOLE_0:def 3;
       suppose a in Edges G;
         then consider xa, ya being set such that xa <> ya and
       A17: xa in Vertices G and ya in Vertices G and
       A18: a = {xa, ya} by Th11;
         per cases by A16,A18,ZFMISC_1:6;
         suppose xa = uG;
          hence contradiction by A17;
         end;
         suppose xa = [y,uG];
          hence contradiction by A17,Th1;
         end;
       end;
       suppose a in A;
         then consider xa, ya being Element of uG such that
       A19: a = {xa,[ya,uG]} and
       A20: {xa,ya} in Edges G;
       A21: xa in uG by A20,Th13;
         per cases by A16,A19,ZFMISC_1:6;
         suppose xa = uG;
          hence contradiction by A21;
         end;
         suppose xa = [y,uG];
          hence contradiction by A21,Th1;
         end;
       end;
   end;
A22: now assume A meets Edges G;
       then consider a being object such that
   A23: a in A and
   A24: a in Edges G by XBOOLE_0:3;
          consider xa, ya being Element of uG such that
       A25: a = {xa,[ya,uG]} and {xa,ya} in Edges G by A23;
          consider xe, ye being set such that xe <> ye and
       A26: xe in Vertices G and
       A27: ye in Vertices G and
       A28: a = {xe, ye} by A24,Th11;
           per cases by A25,A28,ZFMISC_1:6;
         suppose xe = [ya,uG];
          hence contradiction by A26,Th1;
         end;
         suppose ye = [ya,uG];
          hence contradiction by A27,Th1;
         end;
   end;
 thus size Mycielskian G = card ((Edges G) \/ A \/ B) by Th91
   .= card ((Edges G) \/ A) + order G by A11,A13,CARD_2:40
   .= card (Edges G) + 2*(size G) + order G by A12,A22,CARD_2:40
   .= 3*(size G) + order G;
 end;
end;
