
theorem Th34:
  for S1, S2 being non empty RelStr for D being Subset of [:S1,S2
  :] for x being Element of S1, y being Element of S2 holds [x,y] is_<=_than D
  iff x is_<=_than proj1 D & y is_<=_than proj2 D
proof
  let S1, S2 be non empty RelStr, D be Subset of [:S1,S2:], x be Element of S1
  , y be Element of S2;
  set D1 = proj1 D, D2 = proj2 D;
  hereby
    assume
A1: [x,y] is_<=_than D;
    thus x is_<=_than D1
    proof
      let q be Element of S1;
      assume q in D1;
      then consider z being object such that
A2:   [q,z] in D by XTUPLE_0:def 12;
      reconsider d2 = D2 as non empty Subset of S2 by A2,XTUPLE_0:def 13;
      reconsider z as Element of d2 by A2,XTUPLE_0:def 13;
      [x,y] <= [q,z] by A1,A2;
      hence thesis by Th11;
    end;
    thus y is_<=_than D2
    proof
      let q be Element of S2;
      assume q in D2;
      then consider z being object such that
A3:   [z,q] in D by XTUPLE_0:def 13;
      reconsider d1 = D1 as non empty Subset of S1 by A3,XTUPLE_0:def 12;
      reconsider z as Element of d1 by A3,XTUPLE_0:def 12;
      [x,y] <= [z,q] by A1,A3;
      hence thesis by Th11;
    end;
  end;
  assume x is_<=_than proj1 D & y is_<=_than proj2 D;
  then
A4: [x,y] is_<=_than [:D1,D2:] by Th33;
  the carrier of [:S1,S2:] = [:the carrier of S1, the carrier of S2:] by Def2;
  then D c= [:D1,D2:] by Th1;
  hence thesis by A4;
end;
