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 :: (16.1)
  (R~).:B = (R*(R~)).:A
proof
A1: (R*(R~)).:A = R~.:(R.:A) by RELAT_1:126
    .= R~.:(proj2 R) by Th50;
  thus (R~).:B c= (R*(R~)).:A
  proof
    let x be object;
    assume
A2: x in (R~).:B;
A3: (R~).:B = (R~).:(B /\ proj2 R) by Th47;
    (R~).:(B /\ proj2 R) c= R~.:B /\ R~.:(proj2 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~)).:A;
  proj2 R c= rng R;
  then R~.:(proj2 R) c= R~.:B by RELAT_1:123;
  hence thesis by A1,A4;
end;
