
theorem Th38: :: iMR0:
for n being Nat, R being NatRelStr of n, x, y being Nat
 st [x,y] in the InternalRel of Mycielskian R
  holds x < n & y < n or x < n & n <= y & y < 2*n or n <= x & x < 2*n & y < n
     or x = 2*n & n <= y & y < 2*n or n <= x & x < 2*n & y = 2*n
proof
 let n be Nat, R be NatRelStr of n, a, b being Nat;
 set cR = the carrier of R, iR = the InternalRel of R;
 defpred LHS[] means [a,b] in the InternalRel of Mycielskian R;
 defpred RHS[] means
        a < n & b < n or a < n & n <= b & b < 2*n or n <= a & a < 2*n & b < n
     or a = 2*n & n <= b & b < 2*n or n <= a & a < 2*n & b = 2*n;
A1: the InternalRel of Mycielskian R = iR
   \/ { [x,y+n] where x, y is Element of NAT : [x,y] in iR }
   \/ { [x+n,y] where x, y is Element of NAT : [x,y] in iR }
   \/ [: {2*n}, 2*n \ n :] \/ [: 2*n \ n, {2*n} :] by Def9;
 assume A2: LHS[];
   per cases by A2,A1,Th4;
   suppose [a,b] in iR;
     then a in cR & b in cR by ZFMISC_1:87;
     then a in Segm n & b in Segm n by Def7;
     hence RHS[] by NAT_1:44;
   end;
   suppose [a,b] in { [x,y+n] where x, y is Element of NAT : [x,y] in iR };
      then consider x, y being Element of NAT such that
   A3: [x,y+n] = [a,b] and
   A4: [x,y] in iR;
            y in cR by A4,ZFMISC_1:87;
       then y in Segm n by Def7;
       then y < n by NAT_1:44;
       then A5: y+n < n+n by XREAL_1:6;
            x in cR by A4,ZFMISC_1:87;
       then x in Segm n by Def7;
       then x < n by NAT_1:44;
     hence RHS[] by A5,A3,XTUPLE_0:1;
   end;
   suppose [a,b] in { [x+n,y] where x, y is Element of NAT : [x,y] in iR };
      then consider x, y being Element of NAT such that
   A6: [x+n,y] = [a,b] and
   A7: [x,y] in iR;
            x in cR by A7,ZFMISC_1:87;
       then x in Segm n by Def7;
       then x < n by NAT_1:44;
       then A8: x+n < n+n by XREAL_1:6;
            y in cR by A7,ZFMISC_1:87;
       then y in Segm n by Def7;
       then y < n by NAT_1:44;
     hence RHS[] by A8,A6,XTUPLE_0:1;
   end;
   suppose A9: [a,b] in [: {2*n}, 2*n \ n :];
       A10: b in Segm(2*n) \ Segm n by A9,ZFMISC_1:87;
       a in {2*n} by A9,ZFMISC_1:87;
     hence RHS[] by A10,Th2,TARSKI:def 1;
   end;
   suppose A11: [a,b] in [: 2*n \ n, {2*n} :];
       A12: a in Segm(2*n) \ Segm n by A11,ZFMISC_1:87;
       b in {2*n} by A11,ZFMISC_1:87;
     hence RHS[] by A12,Th2,TARSKI:def 1;
   end;
end;
