reserve A,A1,A2,B,B1,B2,C,O for Ordinal,
      R,S for Relation,
      a,b,c,o,l,r for object;
reserve x,y,z,t,r,l for Surreal,
        X,Y,Z for set;

theorem Th33:
  [a,b] in ClosedProd(No_Ord A,A,A) iff a in Day A & b in Day A
proof
  thus [a,b] in ClosedProd(No_Ord A,A,A) implies a in Day A & b in Day A
  by ZFMISC_1:87;
  assume A1:a in Day A & b in Day A;
  then born(No_Ord A,a) c= A & born(No_Ord A,b) c= A by Def8;
  then (born(No_Ord A,a) in A & born(No_Ord A,b) in A)  or
  (born(No_Ord A,a) = A & born(No_Ord A,b) c= A) or
  (born(No_Ord A,a) c= A & born(No_Ord A,b) = A) or
  (born(No_Ord A,a) = A & born(No_Ord A,b) = A)
    by ORDINAL1:11,XBOOLE_0:def 8;
  hence thesis by Def10,A1;
end;
