
theorem Th42: :: iMR1c:
for n being Nat, R being NatRelStr of n, x, y being Nat
 st x in Segm n & [x,y+n] 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 Nat;
 set cR = the carrier of R, iR = the InternalRel of R;
 set MR = Mycielskian R;
 set iMR = the InternalRel of MR;
 assume that
A1: a in Segm n and
A2: [a,b+n] in iMR;
A3: 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 A2,A3,Th4;
    suppose [a,b+n] in iR;
      then b+n in cR by ZFMISC_1:87;
      then b+n in Segm n by Def7;
      then b+n < n by NAT_1:44;
      then b < n-n by XREAL_1:20;
      then b < 0;
      hence [a,b] in iR;
    end;
    suppose [a,b+n] 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
    A4: [a,b+n] = [x,y+n] and
    A5: [x,y] in iR;
        b+n = y+n by A4,XTUPLE_0:1;
      hence [a,b] in iR by A5,A4,XTUPLE_0:1;
    end;
    suppose [a,b+n] 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+n] = [x+n,y] and
    A7: [x,y] in iR;
        b+n = y by A6,XTUPLE_0:1;
        then b+n in cR by A7,ZFMISC_1:87;
        then b+n in Segm n by Def7;
        then b+n < n by NAT_1:44;
        then b < n-n by XREAL_1:20;
        then b < 0;
      hence [a,b] in iR;
    end;
    suppose [a,b+n] in [: {2*n}, 2*n \ n :];
      then consider c, d being object such that
    A8: c in {2*n} and d in 2*n \ n and
    A9: [a,b+n] = [c,d] by ZFMISC_1:def 2;
    A10: c = 2*n by A8,TARSKI:def 1;
    A11: c = a by A9,XTUPLE_0:1;
        n+n < n by A1,A11,A10,NAT_1:44;
        then n < n-n by XREAL_1:20;
        then n < 0;
      hence [a,b] in iR;
    end;
    suppose [a,b+n] in [: 2*n \ n, {2*n} :];
      then consider c, d being object such that
    A12: c in Segm(2*n) \ Segm n and d in {2*n} and
    A13: [a,b+n] = [c,d] by ZFMISC_1:def 2;
        c = a by A13,XTUPLE_0:1;
        then n <= a by A12,Th2;
      hence [a,b] in iR by A1,NAT_1:44;
    end;
end;
