reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];
reserve R for Relation of A,B;
reserve S for Relation of B,C;

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;
