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

theorem Th29:
  A in B implies ClosedProd(R,A,A) = OpenProd(R,A,B)
proof
  assume
  A1: A in B;
  then A2: ClosedProd(R,A,A) c= ClosedProd(R,A,B) by Th17,ORDINAL1:def 2;
  ClosedProd(R,A,B) c= ClosedProd(R,A,A)
  proof
    let x,y be object such that A3:[x,y] in ClosedProd(R,A,B);
    A4:x in Day(R,A) & y in Day(R,A) by A3,ZFMISC_1:87;
    then A5:born(R,x) c= A & born(R,y) c= A by Def8;
    (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 A4,A3,Def10;
    hence thesis by A4,A5,Def10;
  end;
  then A6:ClosedProd(R,A,B) = ClosedProd(R,A,A) by A2,XBOOLE_0:def 10;
  A7:ClosedProd(R,A,B) c= OpenProd(R,A,B) by A1,Th19;
  OpenProd(R,A,B) c= ClosedProd(R,A,B) by Th16;
  hence thesis by A6,A7,XBOOLE_0:def 10;
end;
