reserve A,A1,A2,B,B1,B2,C,O for Ordinal,
      R,S for Relation,
      a,b,c,o,l,r for object;

theorem Th19:
  A in B implies ClosedProd(R,A,B) c= OpenProd(R,A,B)
proof
  assume A1:A in B;
  let x,y be object;
  assume A2:[x,y] in ClosedProd(R,A,B);
  then A3: x in Day(R,A) & y in Day(R,A) by ZFMISC_1:87;
  born(R,x) c= A & born(R,y) c= A by A3,Def8;
  then A4: born(R,x) in B & born(R,y) in B by A1,ORDINAL1:12;
  (born(R,x) in A & born(R,y) in A) or
  (born(R,x) = A & born(R,y) c= B) or
  (born(R,x) c= B & born(R,y) = A) by A3,A2,Def10;
  hence thesis by A3,A4,Def9;
end;
