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 Th50:
  proj1 R = (R~).:B & proj2 R = R.:A
proof
  thus proj1 R = dom R .= rng (R~) by RELAT_1:20
    .= (R~).:B by RELSET_1:22;
  thus proj2 R = rng R .= R.:A by RELSET_1:22;
end;
