
theorem Th41: :: iMR1a:
for n being Nat, R being NatRelStr of n, x, y being Nat
 st [x,y] in the InternalRel of R holds
         [x,y+n] in the InternalRel of Mycielskian R
       & [x+n,y] in the InternalRel of Mycielskian R
proof
  let n be Nat, R be NatRelStr of n, a, b be Nat such that
A1: [a,b] in the InternalRel of R;
 set iR = the InternalRel of R;
 set MR = Mycielskian R;
 set iMR = the InternalRel of MR;
A2: 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;
   reconsider ap1 = a, bp1 = b as Element of NAT by ORDINAL1:def 12;
  [ap1,bp1+n] in { [x,y+n] where x, y is Element of NAT : [x,y] in iR } by A1;
 hence [a,b+n] in iMR by A2,Th4;
  [ap1+n,bp1] in { [x+n,y] where x, y is Element of NAT : [x,y] in iR } by A1;
 hence [a+n,b] in iMR by A2,Th4;
end;
