reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;

theorem Th18:
  N ` A c= N ~ A
proof
A1: N ` A c= A by Th16;
  A c= N ~ A by Th17;
  hence thesis by A1;
end;
