
theorem Th112:
for G being finitely_colorable SimpleGraph
 holds chromatic# Mycielskian G = 1 + chromatic# G
proof :: Mfc1 + contradiction that there is a smaller one
 let G be finitely_colorable SimpleGraph;
  set uG = union G; set MG = Mycielskian G; set uMG = union MG;
  set cnG = chromatic# G; set cnMG = chromatic# MG;
  consider D being Coloring of MG such that
A1: card D = 1+cnG by Th111;
   D is finite by A1; then
A2: cnMG <= 1 + cnG by A1,Def22;
   now
     assume A3: 1+cnG > cnMG;
   A4: cnG >= cnMG by A3,NAT_1:13;
   A5: cnG <= cnMG by Th84,Th68;
   A6: cnG = cnMG by A4,A5,XXREAL_0:1;
       consider E being finite Coloring of MG such that
   A7: card E = cnMG by Def22;
   A8: union E = union MG by EQREL_1:def 4;
   A9: G = MG SubgraphInducedBy uG by Th104;
       reconsider S = uG as Subset of Vertices MG by Th84,ZFMISC_1:77;
       reconsider C = E | S as finite Coloring of G by A9,Th67;
   A10: card C >= cnG by Def22;
   A11: card C <= cnG by A6,A7,MYCIELSK:8;
   A12: card C = cnG by A10,A11,XXREAL_0:1;
   A13: uG in union MG by Th87;
       then consider EuG being set such that
   A14: uG in EuG and
   A15: EuG in E by A8,TARSKI:def 4;
       reconsider EuG as Subset of Vertices MG by A15;
       reconsider uG as Element of Vertices MG by A14,A15;
       set se = EuG /\ S;
   A16: EuG meets S by A15,A6,A7,A12,MYCIELSK:9;
       se in C by A15,A16;
       then consider sev being Element of Vertices G such that
   A17: sev in se and
   A18: for d being Element of C st d <> se
       ex w being Element of Vertices G st w in Adjacent(sev) & w in d
         by A10,A11,Th70,XXREAL_0:1;
   A19: uG is non empty by A16;
        then {[sev,uG]} in MG by Th95;
       then reconsider csev = [sev,uG] as Element of Vertices MG by Th24;
       csev in Vertices MG by A13;
       then csev in union E by EQREL_1:def 4;
       then consider Ecse being set such that
   A20: csev in Ecse and
   A21: Ecse in E by TARSKI:def 4;
       reconsider Ecse as Subset of Vertices MG by A21;
   A22: now assume A23: EuG <> Ecse;
          set sf = Ecse /\ S;
       A24: Ecse meets S by A21,A6,A7,A12,MYCIELSK:9;
       A25: sf in C by A24,A21;
          now assume se = sf;
            then sev in EuG & sev in Ecse by A17,XBOOLE_0:def 4;
            then EuG meets Ecse by XBOOLE_0:3;
           hence contradiction by A23,A21,A15,EQREL_1:def 4;
          end;
          then consider w being Element of Vertices G such that
       A26: w in Adjacent(sev) and
       A27: w in sf by A25,A18;
       A28: w in Ecse by A27,XBOOLE_0:def 4;
       A29: Ecse is stable by A21,Def20;
       A30: csev <> w by A19,Th1;
           {sev, w} in Edges G by A26,Def8;
           then {csev,w} in MG by Th102;
        hence contradiction by A29,A30,A28,A20;
       end;
     A31: {csev,uG} in Edges MG by A19,Th90;
   A32: csev <> uG by Th2;
       EuG is stable by A15,Def20;
     hence contradiction by A32,A31,A22,A20,A14;
   end;
  hence thesis by A2,XXREAL_0:1;
end;
