
theorem Th40: :: iMR1b:
for n being Nat, R being NatRelStr of n, x, y being set
 st x in Segm n & y in Segm n &
  [x,y] in the InternalRel of Mycielskian R
  holds [x,y] in the InternalRel of R
proof
 let n be Nat, R be NatRelStr of n, a, b be set such that
A1: a in Segm n and
A2: b in Segm n and
A3: [a,b] in the InternalRel of Mycielskian R;
 set iR = the InternalRel of R;
 set MR = Mycielskian R;
 set iMR = the InternalRel of MR;
A4: iMR = 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;
  per cases by A3,A4,Th4;
  suppose [a,b] in iR;
    hence [a,b] in iR;
  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
  A5: [a,b] = [x,y+n] and [x,y] in iR;
      b = y+n by A5,XTUPLE_0:1;
      then y+n < n by A2,NAT_1:44;
      then y < n-n by XREAL_1:20;
      then y < 0;
     hence [a,b] in iR;
  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: [a,b] = [x+n,y] and [x,y] in iR;
      a = x+n by A6,XTUPLE_0:1;
      then x+n < n by A1,NAT_1:44;
      then x < n-n by XREAL_1:20;
      then x < 0;
     hence [a,b] in iR;
  end;
  suppose [a,b] in [: {2*n}, 2*n \ n :];
     then consider c, d being object such that
    A7: c in {2*n} and d in 2*n \ n and
    A8: [a,b] = [c,d] by ZFMISC_1:def 2;
    A9: c = 2*n by A7,TARSKI:def 1;
    A10: c = a by A8,XTUPLE_0:1;
        n+n < n by A1,A10,A9,NAT_1:44;
        then n < n-n by XREAL_1:20;
        then n < 0;
      hence [a,b] in iR;
  end;
  suppose [a,b] in [: 2*n \ n, {2*n} :];
     then consider c, d being object such that c in 2*n \ n and
    A11: d in {2*n} and
    A12: [a,b] = [c,d] by ZFMISC_1:def 2;
    A13: d = 2*n by A11,TARSKI:def 1;
    A14: d = b by A12,XTUPLE_0:1;
        n+n < n by A2,A14,A13,NAT_1:44;
        then n < n-n by XREAL_1:20;
        then n < 0;
      hence [a,b] in iR;
  end;
end;
