theorem Th60: :: (20)
  X c= (R~).:^Y iff Y c= R.:^X
proof
  X c= (R~).:^Y iff X misses ((R~).:^Y)` by SUBSET_1:24;
  then X c= (R~).:^Y iff X misses (R~)`.:Y by Th48;
  then
A1: X c= (R~).:^Y iff X /\ (R~)`.:Y = {};
  reconsider S = R` as Relation of A,B;
  X misses S~.:Y iff Y misses S.:X by Th45;
  then X /\ S~.:Y = {} iff Y /\ S.:X = {};
  then X c= (R~).:^Y iff (R.:^X)` /\ Y = {} by A1,Th48,Th59;
  then X c= (R~).:^Y iff (R.:^X)` misses Y;
  hence thesis by SUBSET_1:24;
end;
