
theorem Th46: :: MClique:
for n being Nat, R being irreflexive NatRelStr of n
 st 2 <= clique# R holds clique# R = clique# Mycielskian R
proof
  let n be Nat, R be irreflexive NatRelStr of n such that
A1: 2 <= clique# R and
A2: clique# R <> clique# Mycielskian R;
   set cR = the carrier of R; set iR = the InternalRel of R;
   set MR = Mycielskian R;
   set cMR = the carrier of MR; set iMR = the InternalRel of MR;
   set cnMR = clique# MR;
A3: cR = n by Def7;
A4: cR c= cMR by Th37;
   consider C being finite Clique of R such that
A5: card C = clique# R by DILWORTH:def 4;
     n <= n+n by NAT_1:11;
     then n < 2*n+1 by NAT_1:13;
     then Segm n c= Segm(2*n+1) by NAT_1:39;
     then reconsider S = n as Subset of MR by Def7;
A6: R = subrelstr S by Th45;
   then C is Clique of MR by DILWORTH:28;
   then card C <= cnMR by DILWORTH:def 4;
   then
A7: clique# R < cnMR by A2,A5,XXREAL_0:1;
   then 2 < cnMR by A1,XXREAL_0:2;
   then
A8: 2+1 <= cnMR by NAT_1:13;
   consider D being finite Clique of MR such that
A9: card D = cnMR by DILWORTH:def 4;
   per cases;
   suppose A10: D c= n;
          D /\ S is Clique of R by A6,DILWORTH:29;
     then D is Clique of R by A10,XBOOLE_1:28;
     hence contradiction by A9,A7,DILWORTH:def 4;
   end;
   suppose not D c= n;
     then consider x being object such that
   A11: x in D and
   A12: not x in Segm n;
       x in cMR by A11; then
   A13: x in Segm(2*n+1) by Def7;
       reconsider x as Nat by A13;
       reconsider xp1 = x as Element of MR by A11;
   A14: x >= n by A12,NAT_1:44;
       x < 2*n+1 by A13,NAT_1:44;
       then
   A15: x <= 2*n by NAT_1:13;

   A16: for y being set st y in D & x <> y holds y in Segm n proof
         let y be set such that
       A17: y in D and
       A18: x <> y and
       A19: not y in Segm n;
           y in cMR by A17; then
       A20: y in Segm(2*n+1) by Def7;
           reconsider y as Nat by A20;
           reconsider yp1 = y as Element of MR by A17;
       A21: y >= n by A19,NAT_1:44;
           y < 2*n+1 by A20,NAT_1:44;
           then
       A22: y <= 2*n by NAT_1:13;
           set DD = D \ {x,y};
           {x,y} c= D by A17,A11,ZFMISC_1:32;
           then A23: card DD = card D - card {x,y} by CARD_2:44;
           1+2-2 <= card D -2 by A8,A9,XREAL_1:9;
           then 1 <= card DD by A23,A18,CARD_2:57;
           then consider z being object such that
       A24: z in DD by CARD_1:27,XBOOLE_0:def 1;
      A25: z in D by A24,XBOOLE_0:def 5;
      A26: z in cMR by A24;
           reconsider zp1 = z as Element of MR by A24;
      A27: z in Segm(2*n+1) by A26,Def7;
           reconsider z as Nat by A27;
           x in {x,y} by TARSKI:def 2;
           then
      A28: z <> x by A24,XBOOLE_0:def 5;
           y in {x,y} by TARSKI:def 2;
           then
      A29: z <> y by A24,XBOOLE_0:def 5;
         per cases by A15,A22,XXREAL_0:1;
         suppose A30: x < 2*n & y < 2*n;
           xp1 <= yp1 or yp1 <= xp1 by A11,A17,A18,DILWORTH:6;
           then [x,y] in iMR or [y,x] in iMR by ORDERS_2:def 5;
          hence contradiction by A14,A30,A21,Th38;
         end;
         suppose A31: x < 2*n & y = 2*n;
            xp1 <= zp1 or zp1 <= xp1 by A28,A25,A11,DILWORTH:6;
            then A32: [x,z] in iMR or [z,x] in iMR by ORDERS_2:def 5;
            yp1 <= zp1 or zp1 <= yp1 by A29,A25,A17,DILWORTH:6;
            then [y,z] in iMR or [z,y] in iMR by ORDERS_2:def 5;
            then n <= z & z < 2*n by A31,A21,Th38;
          hence contradiction by A32,A31,A14,Th38;
         end;
         suppose A33: x = 2*n & y < 2*n;
            yp1 <= zp1 or zp1 <= yp1 by A29,A25,A17,DILWORTH:6;
            then A34: [y,z] in iMR or [z,y] in iMR by ORDERS_2:def 5;
            xp1 <= zp1 or zp1 <= xp1 by A28,A25,A11,DILWORTH:6;
            then [x,z] in iMR or [z,x] in iMR by ORDERS_2:def 5;
            then n <= z & z < 2*n by A33,A14,Th38;
          hence contradiction by A34,A33,A21,Th38;
         end;
         suppose x = 2*n & y = 2*n;
          hence contradiction by A18;
         end;
       end;
  A35: card (D\{x}) = card D - card {x} by A11,EULER_1:4
                       .= card D - 1 by CARD_1:30;
       per cases by A15,XXREAL_0:1;
       suppose A36: x < 2*n;
         consider xx being Nat such that
   A37: x = n+xx by A14,NAT_1:10;
           n + xx < n + n by A36,A37;
           then A38: xx < n by XREAL_1:6;
           then A39: xx in Segm n by NAT_1:44;
           reconsider xxp1 = xx as Element of MR by A39,A4,A3;
       A40: now
            assume xx in D;
              then xp1 <= xxp1 or xxp1 <= xp1 by A11,A38,A14,DILWORTH:6;
              then [x,xx] in iMR or [xx,x] in iMR by ORDERS_2:def 5;
              then [xx,xx] in iR or [xx,xx] in iR by A39,A37,Th42,Th43;
            hence contradiction by A39,A3,NECKLACE:def 5;
           end;
           set DD = (D\{x}) \/ {xx};
           DD c= cR proof
             let a be object;
             assume a in DD;
             then a in D\{x} or a in {xx} by XBOOLE_0:def 3;
             then a in D & not a in {x} or a = xx
                         by TARSKI:def 1,XBOOLE_0:def 5;
             then a in D & a <> x or a = xx by TARSKI:def 1;
             hence a in cR by A38,A16,A3,NAT_1:44;
           end;
           then reconsider DD as Subset of R;
           now
             let a, b be Element of R such that
           A41: a in DD and
           A42: b in DD and
           A43: a <> b;
               a in D\{x} or a in {xx} by A41,XBOOLE_0:def 3;
               then
           A44
: a in D & not a in {x} or a = xx by TARSKI:def 1,XBOOLE_0:def 5;
               b in D\{x} or b in {xx} by A42,XBOOLE_0:def 3;
               then
           A45
: b in D & not b in {x} or b = xx by TARSKI:def 1,XBOOLE_0:def 5;
           A46: a in cR & b in cR by A41;
              then a in Segm n & b in Segm n by A3;
              then reconsider an = a, bn = b as Nat;
               reconsider ap1 = a, bp1 = b as Element of MR by A46,A4;
             per cases by A43,A44,A45,TARSKI:def 1;
             suppose A47: a in D & a <> x & b in D & b <> x;
                    ap1 <= bp1 or bp1 <= ap1 by A47,A43,DILWORTH:6;
               hence a <= b or b <= a by A6,A41,YELLOW_0:60;
             end;
             suppose A48: a in D & a <> x & b = xx;
                    ap1 <= xp1 or xp1 <= ap1 by A48,A11,DILWORTH:6;
               then [ap1,x] in iMR or [x,ap1] in iMR by ORDERS_2:def 5;
               then [an, xx] in iR or [xx,an] in iR
                  by A3,A37,Th42,Th43,A39;
               hence a <= b or b <= a by A48,ORDERS_2:def 5;
             end;
             suppose A49: a = xx & b in D & b <> x;
                    bp1 <= xp1 or xp1 <= bp1 by A49,A11,DILWORTH:6;
               then [bp1,x] in iMR or [x,bp1] in iMR by ORDERS_2:def 5;
               then [bn, xx] in iR or [xx,bn] in iR
                  by A3,A37,Th42,Th43,A39;
               hence a <= b or b <= a by A49,ORDERS_2:def 5;
             end;
           end;
           then reconsider DD as Clique of R by DILWORTH:6;
       A50: not xx in D\{x} by A40,XBOOLE_0:def 5;
            card DD = card D - 1 + 1 by A50,A35,CARD_2:41 .= card D;
         hence contradiction by A9,A7,DILWORTH:def 4;
       end;
       suppose A51: x = 2*n;
       2+1-1 <= card D -1 by A9,A8,XREAL_1:9;
       then Segm 2 c= Segm card (D\{x}) by A35,NAT_1:39;
       then consider y, z being object such that
   A52: y in D\{x} and z in D\{x} and y <> z by PENCIL_1:2;
   A53: y in D by A52,ZFMISC_1:56;
   A54: x <> y by A52,ZFMISC_1:56;
       y in the carrier of MR by A52;
       then y in Segm(2*n+1) by Def7;
       then reconsider y as Nat;
       reconsider yp1 = y as Element of MR by A52;
       yp1 <= xp1 or xp1 <= yp1 by A54,A53,A11,DILWORTH:6;
       then  A55: [y,x] in iMR or [x,y] in iMR by ORDERS_2:def 5;
       y in Segm n by A16,A53,A54;
       then  A56: y < n by NAT_1:44;
          n <= n+n by NAT_1:11;
        hence contradiction by A55,A51,A56,Th38;
       end;
   end;
end;
