
theorem Th109: :: MClique2
for G being with_finite_clique# SimpleGraph st 2 <= clique# G
for D being finite Clique of Mycielskian G holds order D <= clique# G
proof
 let G be with_finite_clique# SimpleGraph such that
A1: 2 <= clique# G;
 let D be finite Clique of Mycielskian G such that
A2: order D > clique# G;
   set MG = Mycielskian G, uG = union G;
A3: Vertices D c= Vertices MG by ZFMISC_1:77;
A4: Edges D c= Edges MG by Th14;
   2 < order D by A2,A1,XXREAL_0:2;
then A5: 2+1 <= order D by NAT_1:13;
   per cases;
   suppose D c= G;
    hence contradiction by A2,Def15;
   end;
   suppose not D c= G;
          then consider e being object such that
      A6: e in D and
      A7: not e in G;
     now
      assume A8: Vertices D c= Vertices G;
      A9: e <> {} by A7,Th20;
          now assume not e in Edges D;
          then consider y being set such that
      A10: e = {y} and
      A11: y in Vertices D by A9,A6,Th29;
          thus contradiction by A7,A10,A11,A8,Th24;
        end;
        then consider x, y being set such that x <> y and
      A12: x in Vertices D and
      A13: y in Vertices D and
      A14: e = {x, y} by Th11;
      thus contradiction by A6,A8,A14,A7,Th103,A12,A13;
    end;
    then consider v being object such that
   A15: v in Vertices D and
   A16: not v in Vertices G;
        Segm 3 c= Segm order D by A5,NAT_1:39;
        then consider v1, v2 being object such that
   A17: v1 in Vertices D and
   A18: v2 in Vertices D and
   A19: v1<>v and
   A20: v2<>v and
   A21: v1<>v2 by Th5;
        {v,v1} in D by A15,A17,Th53; then
   A22: {v, v1} in Edges D by A19,Th12;
        {v,v2} in D by A15,A18,Th53; then
   A23: {v, v2} in Edges D by A20,Th12;
        {v1,v2} in D by A17,A18,Th53; then
   A24: {v1, v2} in Edges D by A21,Th12;
    per cases by A15,A3,A16,Th85;
    suppose A25: v = uG;
        consider x being object such that x in union G and
   A26: v1 = [x, union G] by A25,A22,A4,Th94;
        consider y being object such that y in union G and
   A27: v2 = [y, union G] by A25,A23,A4,Th94;
      thus contradiction by A24,A4,A26,A27,Th97;
    end;
    suppose ex x being set st x in union G & v = [x,union G];
      then consider x being set such that
    A28: x in uG and
    A29: v = [x,uG];
    set E = D SubgraphInducedBy union G;
    reconsider F = G SubgraphInducedBy ({x} \/ union E) as finite SimpleGraph;
    A30: Vertices F = {x} \/ Vertices E proof
         thus Vertices F c= {x} \/ Vertices E proof
           let a be object;
           assume a in Vertices F;
            then a in (union G) /\ ({x} \/ union E) by Th45;
then     A31: a in {x} \/ union E by XBOOLE_0:def 4;
           per cases by A31,XBOOLE_0:def 3;
           suppose a in {x};
            hence thesis by XBOOLE_0:def 3;
           end;
           suppose a in union E;
            hence thesis by XBOOLE_0:def 3;
           end;
         end;
         let a be object;
         assume A32: a in {x} \/ Vertices E;
         per cases by A32,XBOOLE_0:def 3;
         suppose a in {x};
then     A33: a = x by TARSKI:def 1;
           x in {x} by TARSKI:def 1;
then       x in {x} \/ union E by XBOOLE_0:def 3;
then       x in (union G) /\ ({x} \/ union E) by A28,XBOOLE_0:def 4;
          hence a in Vertices F by A33,Th45;
         end;
         suppose a in Vertices E;
then         a in (union D) /\ union G by Th45;
then         a in union G by XBOOLE_0:def 4;
then         a in (union G) /\ ({x} \/ union E) by A32,XBOOLE_0:def 4;
           hence a in Vertices F by Th45;
         end;
         end;
    A34: union E c= union D by ZFMISC_1:77;
    A35: now
           assume x in union E;
            then {[x,uG],x} in D by A34,A15,A29,Th53;
           hence contradiction by Th100;
         end;
         reconsider Fs = F as SimpleGraph-like Subset of G;
       now
         let a, b be set such that
       A36: a <> b and
       A37: a in union Fs and
       A38: b in union Fs;
       A39:  a in (union G) /\ ({x} \/ union E) by A37,Th45;
  then A40: a in union G & a in {x} \/ union E by XBOOLE_0:def 4;
       A41: b in (union G) /\ ({x} \/ union E) by A38,Th45;
  then A42: b in union G & b in {x} \/ union E by XBOOLE_0:def 4;
       A43: a in Vertices G by A39,XBOOLE_0:def 4;
       A44: b in Vertices G by A41,XBOOLE_0:def 4;
            x in {x} by TARSKI:def 1;
   then A45: x in {x} \/ union E by XBOOLE_0:def 3;
        {a,b} in Fs proof
         per cases by A40,A42,XBOOLE_0:def 3;
         suppose a in {x} & b in {x};
           then A46: a = x & b = x by TARSKI:def 1;
           then {a,b} = {x} by ENUMSET1:29;
          hence {a,b} in Fs by A46,A37,Th24;
         end;
         suppose A47: a in {x} & b in union E;
    then A48: a = x by TARSKI:def 1;
             b in (union D) /\ union G by A47,Th45;
    then A49: b in union D & b in uG by XBOOLE_0:def 4;
    then     {[x,uG], b} in D by A15,A29,Th53;
    then     {x,b} in G by A49,Th101;
         hence {a,b} in Fs by A45,A48,A42,Lm10;
         end;
         suppose A50: b in {x} & a in union E;
    then A51: b = x by TARSKI:def 1;
             a in (union D) /\ union G by A50,Th45;
    then A52: a in union D & a in uG by XBOOLE_0:def 4;
    then     {[x,uG], a} in D by A15,A29,Th53;
    then     {x,a} in G by A52,Th101;
         hence {a,b} in Fs by A45,A51,A40,Lm10;
         end;
         suppose a in union E & b in union E;
           then a in (union D) /\ union G & b in (union D) /\ union G by Th45;
           then a in union D & b in union D by XBOOLE_0:def 4;
           then {a,b} in D by Th53;
           then {a,b} in G by A43,A44,Th103;
          hence {a,b} in Fs by A40,A42,Lm10;
         end;
         end;
         hence {a,b} in Edges Fs by A36,Th12;
        end; then
    A53: Fs is clique by Th47;
    A54: Vertices D c= {v} \/ Vertices E proof
          let a be object such that
      A55: a in Vertices D;
          per cases;
          suppose a = v;
            then a in {v} by TARSKI:def 1;
            hence thesis by XBOOLE_0:def 3;
          end;
          suppose A56: a <> v;
               {a,[x,uG]} in D by A29,A15,A55,Th53; then
               {a,[x,uG]} in Edges D by A56,A29,Th12;
               then a in uG or a = uG by A4,Th99;
               then a in Vertices E by A5,Th105,A55,Lm8;
            hence thesis by XBOOLE_0:def 3;
          end;
        end;
    A57: Vertices E c= Vertices D by ZFMISC_1:77;
    A58: {v} \/ Vertices E c= Vertices D proof
          let a be object;
          assume a in {v} \/ Vertices E;
          then a in {v} or a in Vertices E by XBOOLE_0:def 3;
          hence thesis by A15,A57,TARSKI:def 1;
         end;
    A59: not v in Vertices E by A29,Lm7,Th1;
       order F = 1 + card Vertices E by A30,A35,CARD_2:41
              .= card ({v} \/ Vertices E) by A59,CARD_2:41
              .= order D by A58,A54,XBOOLE_0:def 10;
      hence contradiction by A2,A53,Def15;
    end;
   end;
end;
