reserve U for Universe;

theorem Th2:
  the InternalRel of Necklace 4 = {[0,1],[1,0],[1,2],[2,1],[2,3],[3 ,2]}
proof
  set A = {[0,1],[1,0],[1,2],[2,1],[2,3],[3,2]}, B = the InternalRel of
  Necklace 4;
A1: [0+1,0] in {[i+1,i] where i is Element of NAT:i+1 < 4};
A2: B = {[i,i+1] where i is Element of NAT:i+1 < 4} \/ {[i+1,i] where i is
  Element of NAT:i+1 < 4} by NECKLACE:18;
A3: B c= A
  proof
    let x be object;
    assume
A4: x in B;
    then consider y,z being object such that
A5: x = [y,z] by RELAT_1:def 1;
    per cases by A2,A4,XBOOLE_0:def 3;
    suppose
      x in {[i,i+1] where i is Element of NAT: i+1<4};
      then consider i be Element of NAT such that
A6:   [y,z] = [i,i+1] and
A7:   i+1 < 4 by A5;
A8:   y = i by A6,XTUPLE_0:1;
      i + 1 < 1 + 3 by A7;
      then i < 2+1 by XREAL_1:7;
      then i <= 2 by NAT_1:13;
      then y = 0 or ... or y = 2 by A8;
      hence thesis by A5,A6,A8,ENUMSET1:def 4;
    end;
    suppose
      x in {[i+1,i] where i is Element of NAT: i+1<4};
      then consider i be Element of NAT such that
A9:   [y,z] = [i+1,i] and
A10:  i+1 < 4 by A5;
A11:  z = i by A9,XTUPLE_0:1;
      i + 1 < 1 + 3 by A10;
      then i < 2+1 by XREAL_1:7;
      then i <= 2 by NAT_1:13;
      then z = 0 or ... or z = 2 by A11;
      hence thesis by A5,A9,A11,ENUMSET1:def 4;
    end;
  end;
A12: [2+1,1+1] in {[i+1,i] where i is Element of NAT:i+1 < 4};
A13: [1+1,2+1] in {[i,i+1] where i is Element of NAT:i+1 < 4};
A14: [1+1,0+1] in {[i+1,i] where i is Element of NAT:i+1 < 4};
A15: [0+1,1+1] in {[i,i+1] where i is Element of NAT:i+1 < 4};
A16: [0,0+1] in {[i,i+1] where i is Element of NAT:i+1 < 4};
  A c= B
  proof
    let x be object;
    assume
A17: x in A;
    per cases by A17,ENUMSET1:def 4;
    suppose
      x = [0,1];
      hence thesis by A2,A16,XBOOLE_0:def 3;
    end;
    suppose
      x=[1,0];
      hence thesis by A2,A1,XBOOLE_0:def 3;
    end;
    suppose
      x=[1,2];
      hence thesis by A2,A15,XBOOLE_0:def 3;
    end;
    suppose
      x=[2,1];
      hence thesis by A2,A14,XBOOLE_0:def 3;
    end;
    suppose
      x=[2,3];
      hence thesis by A2,A13,XBOOLE_0:def 3;
    end;
    suppose
      x=[3,2];
      hence thesis by A2,A12,XBOOLE_0:def 3;
    end;
  end;
  hence thesis by A3,XBOOLE_0:def 10;
end;
