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 :: (23)
  R.:^X = R.:^( R~.:^(R.:^X) ) & R~.:^Y = (R~).:^(R.:^((R~).:^Y))
proof
A1: R.:^X c= R.:^( R~.:^(R.:^X) ) by Th60;
  X c= (R~).:^(R.:^X) by Th60;
  then
  R.:^( R~.:^(R.:^X) ) c= R.:^X by Th31;
  hence R.:^X = R.:^( R~.:^(R.:^X) ) by A1;
  thus R~.:^Y c= (R~).:^(R.:^((R~).:^Y)) by Th60;
  Y c= R.:^((R~).:^Y) by Th60;
  hence thesis by Th31;
end;
