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 :: (16.2)
  R.:A = (R~*R).:B
proof
A1: (R~*R).:B = R.:(R~.:B) by RELAT_1:126
    .= R.:(proj1 R) by Th50;
  thus R.:A c= (R~*R).:B
  proof
    let x be object;
    assume
A2: x in R.:A;
A3: R.:A = R.:(A /\ proj1 R) by Th46;
    R.:(A /\ proj1 R) c= R.:A /\ R.:(proj1 R) by RELAT_1:121;
    hence thesis by A1,A2,A3,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A4: x in (R~*R).:B;
  proj1 R c= dom R;
  then R.:(proj1 R) c= R.:A by RELAT_1:123;
  hence thesis by A1,A4;
end;
