reserve A,B,a,b,c,d,e,f,g,h for set;

theorem Th11:
  the InternalRel of ComplRelStr Necklace 4 = {[0,2],[2,0],[0,3],[
  3,0],[1,3],[3,1]}
proof
  set N4 = Necklace 4, cN4 = the carrier of N4, CmpN4 = ComplRelStr N4;
A1: the carrier of Necklace 4 = {0,1,2,3} by NECKLACE:1,20;
  thus the InternalRel of CmpN4 c= {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]}
  proof
    let x be object;
    assume x in the InternalRel of CmpN4;
    then
A2: x in (the InternalRel of N4)` \ id cN4 by NECKLACE:def 8;
    then
A3: not x in id cN4 by XBOOLE_0:def 5;
    x in (the InternalRel of N4)` by A2,XBOOLE_0:def 5;
    then
A4: x in [:cN4,cN4:] \ (the InternalRel of N4) by SUBSET_1:def 4;
    consider a1,b1 being object such that
A5: a1 in cN4 and
A6: b1 in cN4 and
A7: x = [a1,b1] by A2,ZFMISC_1:def 2;
    per cases by A1,A5,A6,ENUMSET1:def 2;
    suppose
      a1 = 0 & b1 = 0;
      hence thesis by A3,A5,A7,RELAT_1:def 10;
    end;
    suppose
      a1 = 0 & b1 = 1;
      then x in the InternalRel of N4 by A7,ENUMSET1:def 4,NECKLA_2:2;
      hence thesis by A4,XBOOLE_0:def 5;
    end;
    suppose
      a1 = 0 & b1 = 2;
      hence thesis by A7,ENUMSET1:def 4;
    end;
    suppose
      a1 = 0 & b1 = 3;
      hence thesis by A7,ENUMSET1:def 4;
    end;
    suppose
      a1 = 1 & b1 = 0;
      then x in the InternalRel of N4 by A7,ENUMSET1:def 4,NECKLA_2:2;
      hence thesis by A4,XBOOLE_0:def 5;
    end;
    suppose
      a1 = 2 & b1 = 0;
      hence thesis by A7,ENUMSET1:def 4;
    end;
    suppose
      a1 = 3 & b1 = 0;
      hence thesis by A7,ENUMSET1:def 4;
    end;
    suppose
      a1 = 1 & b1 = 1;
      hence thesis by A3,A5,A7,RELAT_1:def 10;
    end;
    suppose
      a1 = 1 & b1 = 2;
      then x in the InternalRel of N4 by A7,ENUMSET1:def 4,NECKLA_2:2;
      hence thesis by A4,XBOOLE_0:def 5;
    end;
    suppose
      a1 = 1 & b1 = 3;
      hence thesis by A7,ENUMSET1:def 4;
    end;
    suppose
      a1 = 2 & b1 = 2;
      hence thesis by A3,A5,A7,RELAT_1:def 10;
    end;
    suppose
      a1 = 2 & b1 = 1;
      then x in the InternalRel of N4 by A7,ENUMSET1:def 4,NECKLA_2:2;
      hence thesis by A4,XBOOLE_0:def 5;
    end;
    suppose
      a1 = 2 & b1 = 3;
      then x in the InternalRel of N4 by A7,ENUMSET1:def 4,NECKLA_2:2;
      hence thesis by A4,XBOOLE_0:def 5;
    end;
    suppose
      a1 = 3 & b1 = 3;
      hence thesis by A3,A5,A7,RELAT_1:def 10;
    end;
    suppose
      a1 = 3 & b1 = 1;
      hence thesis by A7,ENUMSET1:def 4;
    end;
    suppose
      a1 = 3 & b1 = 2;
      then x in the InternalRel of N4 by A7,ENUMSET1:def 4,NECKLA_2:2;
      hence thesis by A4,XBOOLE_0:def 5;
    end;
  end;
  let a be object;
  assume
A8: a in {[0,2],[2,0],[0,3],[3,0],[1,3],[3,1]};
  per cases by A8,ENUMSET1:def 4;
  suppose
A9: a = [0,2];
A10: not a in the InternalRel of N4
    proof
      assume
A11:  not thesis;
      per cases by A11,ENUMSET1:def 4,NECKLA_2:2;
      suppose
        a = [0,1];
        hence contradiction by A9,XTUPLE_0:1;
      end;
      suppose
        a = [1,0];
        hence contradiction by A9,XTUPLE_0:1;
      end;
      suppose
        a = [1,2];
        hence contradiction by A9,XTUPLE_0:1;
      end;
      suppose
        a = [2,1];
        hence contradiction by A9,XTUPLE_0:1;
      end;
      suppose
        a = [2,3];
        hence contradiction by A9,XTUPLE_0:1;
      end;
      suppose
        a = [3,2];
        hence contradiction by A9,XTUPLE_0:1;
      end;
    end;
    0 in cN4 & 2 in cN4 by A1,ENUMSET1:def 2;
    then a in [:cN4,cN4:] by A9,ZFMISC_1:87;
    then a in [:cN4,cN4:]\ (the InternalRel of N4) by A10,XBOOLE_0:def 5;
    then
A12: a in (the InternalRel of N4)` by SUBSET_1:def 4;
    not a in id cN4 by A9,RELAT_1:def 10;
    then a in (the InternalRel of N4)` \ id cN4 by A12,XBOOLE_0:def 5;
    hence thesis by NECKLACE:def 8;
  end;
  suppose
A13: a = [2,0];
A14: not a in the InternalRel of N4
    proof
      assume
A15:  not thesis;
      per cases by A15,ENUMSET1:def 4,NECKLA_2:2;
      suppose
        a = [0,1];
        hence contradiction by A13,XTUPLE_0:1;
      end;
      suppose
        a = [1,0];
        hence contradiction by A13,XTUPLE_0:1;
      end;
      suppose
        a = [1,2];
        hence contradiction by A13,XTUPLE_0:1;
      end;
      suppose
        a = [2,1];
        hence contradiction by A13,XTUPLE_0:1;
      end;
      suppose
        a = [2,3];
        hence contradiction by A13,XTUPLE_0:1;
      end;
      suppose
        a = [3,2];
        hence contradiction by A13,XTUPLE_0:1;
      end;
    end;
    0 in cN4 & 2 in cN4 by A1,ENUMSET1:def 2;
    then a in [:cN4,cN4:] by A13,ZFMISC_1:87;
    then a in [:cN4,cN4:]\ (the InternalRel of N4) by A14,XBOOLE_0:def 5;
    then
A16: a in (the InternalRel of N4)` by SUBSET_1:def 4;
    not a in id cN4 by A13,RELAT_1:def 10;
    then a in (the InternalRel of N4)` \ id cN4 by A16,XBOOLE_0:def 5;
    hence thesis by NECKLACE:def 8;
  end;
  suppose
A17: a = [0,3];
A18: not a in the InternalRel of N4
    proof
      assume
A19:  not thesis;
      per cases by A19,ENUMSET1:def 4,NECKLA_2:2;
      suppose
        a = [0,1];
        hence contradiction by A17,XTUPLE_0:1;
      end;
      suppose
        a = [1,0];
        hence contradiction by A17,XTUPLE_0:1;
      end;
      suppose
        a = [1,2];
        hence contradiction by A17,XTUPLE_0:1;
      end;
      suppose
        a = [2,1];
        hence contradiction by A17,XTUPLE_0:1;
      end;
      suppose
        a = [2,3];
        hence contradiction by A17,XTUPLE_0:1;
      end;
      suppose
        a = [3,2];
        hence contradiction by A17,XTUPLE_0:1;
      end;
    end;
    0 in cN4 & 3 in cN4 by A1,ENUMSET1:def 2;
    then a in [:cN4,cN4:] by A17,ZFMISC_1:87;
    then a in [:cN4,cN4:]\ (the InternalRel of N4) by A18,XBOOLE_0:def 5;
    then
A20: a in (the InternalRel of N4)` by SUBSET_1:def 4;
    not a in id cN4 by A17,RELAT_1:def 10;
    then a in (the InternalRel of N4)` \ id cN4 by A20,XBOOLE_0:def 5;
    hence thesis by NECKLACE:def 8;
  end;
  suppose
A21: a = [3,0];
A22: not a in the InternalRel of N4
    proof
      assume
A23:  not thesis;
      per cases by A23,ENUMSET1:def 4,NECKLA_2:2;
      suppose
        a = [0,1];
        hence contradiction by A21,XTUPLE_0:1;
      end;
      suppose
        a = [1,0];
        hence contradiction by A21,XTUPLE_0:1;
      end;
      suppose
        a = [1,2];
        hence contradiction by A21,XTUPLE_0:1;
      end;
      suppose
        a = [2,1];
        hence contradiction by A21,XTUPLE_0:1;
      end;
      suppose
        a = [2,3];
        hence contradiction by A21,XTUPLE_0:1;
      end;
      suppose
        a = [3,2];
        hence contradiction by A21,XTUPLE_0:1;
      end;
    end;
    0 in cN4 & 3 in cN4 by A1,ENUMSET1:def 2;
    then a in [:cN4,cN4:] by A21,ZFMISC_1:87;
    then a in [:cN4,cN4:]\ (the InternalRel of N4) by A22,XBOOLE_0:def 5;
    then
A24: a in (the InternalRel of N4)` by SUBSET_1:def 4;
    not a in id cN4 by A21,RELAT_1:def 10;
    then a in (the InternalRel of N4)` \ id cN4 by A24,XBOOLE_0:def 5;
    hence thesis by NECKLACE:def 8;
  end;
  suppose
A25: a = [1,3];
A26: not a in the InternalRel of N4
    proof
      assume
A27:  not thesis;
      per cases by A27,ENUMSET1:def 4,NECKLA_2:2;
      suppose
        a = [0,1];
        hence contradiction by A25,XTUPLE_0:1;
      end;
      suppose
        a = [1,0];
        hence contradiction by A25,XTUPLE_0:1;
      end;
      suppose
        a = [1,2];
        hence contradiction by A25,XTUPLE_0:1;
      end;
      suppose
        a = [2,1];
        hence contradiction by A25,XTUPLE_0:1;
      end;
      suppose
        a = [2,3];
        hence contradiction by A25,XTUPLE_0:1;
      end;
      suppose
        a = [3,2];
        hence contradiction by A25,XTUPLE_0:1;
      end;
    end;
    1 in cN4 & 3 in cN4 by A1,ENUMSET1:def 2;
    then a in [:cN4,cN4:] by A25,ZFMISC_1:87;
    then a in [:cN4,cN4:]\ (the InternalRel of N4) by A26,XBOOLE_0:def 5;
    then
A28: a in (the InternalRel of N4)` by SUBSET_1:def 4;
    not a in id cN4 by A25,RELAT_1:def 10;
    then a in (the InternalRel of N4)` \ id cN4 by A28,XBOOLE_0:def 5;
    hence thesis by NECKLACE:def 8;
  end;
  suppose
A29: a = [3,1];
A30: not a in the InternalRel of N4
    proof
      assume
A31:  not thesis;
      per cases by A31,ENUMSET1:def 4,NECKLA_2:2;
      suppose
        a = [0,1];
        hence contradiction by A29,XTUPLE_0:1;
      end;
      suppose
        a = [1,0];
        hence contradiction by A29,XTUPLE_0:1;
      end;
      suppose
        a = [1,2];
        hence contradiction by A29,XTUPLE_0:1;
      end;
      suppose
        a = [2,1];
        hence contradiction by A29,XTUPLE_0:1;
      end;
      suppose
        a = [2,3];
        hence contradiction by A29,XTUPLE_0:1;
      end;
      suppose
        a = [3,2];
        hence contradiction by A29,XTUPLE_0:1;
      end;
    end;
    1 in cN4 & 3 in cN4 by A1,ENUMSET1:def 2;
    then a in [:cN4,cN4:] by A29,ZFMISC_1:87;
    then a in [:cN4,cN4:]\ (the InternalRel of N4) by A30,XBOOLE_0:def 5;
    then
A32: a in (the InternalRel of N4)` by SUBSET_1:def 4;
    not a in id cN4 by A29,RELAT_1:def 10;
    then a in (the InternalRel of N4)` \ id cN4 by A32,XBOOLE_0:def 5;
    hence thesis by NECKLACE:def 8;
  end;
end;
