
theorem Th43: :: iMR1d:
for n being Nat, R being NatRelStr of n, x, y being Nat
 st y in Segm n & [x+n,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 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: b in Segm n and
A2: [a+n,b] 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+n,b] in iR;
      then a+n in cR by ZFMISC_1:87;
      then a+n in Segm n by Def7;
      then a+n < n by NAT_1:44;
      then a < n-n by XREAL_1:20;
      then a < 0;
      hence [a,b] in iR;
    end;
    suppose [a+n,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
    A4: [a+n,b] = [x,y+n] and
    A5: [x,y] in iR;
        a+n = x by A4,XTUPLE_0:1;
        then a+n in cR by A5,ZFMISC_1:87;
        then a+n in Segm n by Def7;
        then a+n < n by NAT_1:44;
        then a < n-n by XREAL_1:20;
        then a < 0;
      hence [a,b] in iR;
    end;
    suppose [a+n,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+n,b] = [x+n,y] and
    A7: [x,y] in iR;
        a+n = x+n by A6,XTUPLE_0:1;
      hence [a,b] in iR by A7,A6,XTUPLE_0:1;
    end;
    suppose [a+n,b] in [: {2*n}, 2*n \ n :];
      then consider c, d being object such that c in {2*n} and
    A8: d in Segm(2*n) \ Segm n and
    A9: [a+n,b] = [c,d] by ZFMISC_1:def 2;
        b = d by A9,XTUPLE_0:1;
        then n <= b by A8,Th2;
      hence [a,b] in iR by A1,NAT_1:44;
    end;
    suppose [a+n,b] in [: 2*n \ n, {2*n} :];
      then consider c, d being object such that c in 2*n \ n and
    A10: d in {2*n} and
    A11: [a+n,b] = [c,d] by ZFMISC_1:def 2;
    A12: d = 2*n by A10,TARSKI:def 1;
        d = b by A11,XTUPLE_0:1;
        then n+n < n by A1,A12,NAT_1:44;
        then n < n-n by XREAL_1:20;
        then n < 0;
      hence [a,b] in iR;
    end;
end;
