
theorem Th44: :: iMR1e:
for n being Nat, R being NatRelStr of n, m being Nat
 st n <= m & m < 2*n
  holds [m,2*n] in the InternalRel of Mycielskian R
      & [2*n,m] in the InternalRel of Mycielskian R
proof
 let n be Nat, R be NatRelStr of n, m be Nat such that
A1: n <= m and
A2: m < 2*n;
 set iR = the InternalRel of R;
 set MR = Mycielskian R;
 set iMR = the InternalRel of MR;
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;
A4: m in Segm(2*n) \ Segm n by A1,A2,Th2;
A5: 2*n in {2*n} by TARSKI:def 1;
   then [m,2*n] in [: 2*n \ n, {2*n} :] by A4,ZFMISC_1:def 2;
  hence [m,2*n] in iMR by A3,XBOOLE_0:def 3;
        [2*n,m] in [: {2*n}, 2*n \ n :] by A4,A5,ZFMISC_1:def 2;
  hence [2*n,m] in iMR by A3,Th4;
end;
