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

theorem Th17:
  A1 in A2 or (A1=A2 & B1 c= B2) implies
    ClosedProd(R,A1,B1) c= ClosedProd(R,A2,B2)
proof
  assume A1: A1 in A2 or (A1=A2 & B1 c= B2);
  A2:A1 c= A2 by A1,ORDINAL1:def 2;
  A3:Day(R,A1) c= Day(R,A2) by Th9,A1,ORDINAL1:def 2;
  let x,y be object such that A4:[x,y] in ClosedProd(R,A1,B1);
  A5:x in Day(R,A1) & y in Day(R,A1) by A4,ZFMISC_1:87;
  per cases by A5,A4,Def10;
  suppose (born(R,x) in A1 & born(R,y) in A1);
    hence thesis by A2,A3,A5,Def10;
  end;
  suppose A6:born(R,x) = A1 & born(R,y) c= B1;
    per cases by A1;
    suppose A7: A1 in A2;
      born(R,y) c= A1 by A5,Def8;
      then born(R,y) in A2 by A7,ORDINAL1:12;
      hence thesis by A6,A7,A5,A3,Def10;
    end;
    suppose
      A1=A2 & B1 c= B2;
      then born(R,x) = A2 & born(R,y) c= B2 by A6,XBOOLE_1:1;
      hence thesis by A5,A3,Def10;
    end;
  end;
  suppose A8:born(R,x) c= B1 & born(R,y) = A1;
    per cases by A1;
    suppose A9: A1 in A2;
      born(R,x) c= A1 by A5,Def8;
      then born(R,x) in A2 by A9,ORDINAL1:12;
      hence thesis by A8,A9,A5,A3,Def10;
    end;
    suppose
      A1=A2 & B1 c= B2;
      then born(R,y) = A2 & born(R,x) c= B2 by A8,XBOOLE_1:1;
      hence thesis by A5,A3,Def10;
    end;
  end;
end;
