reserve x,y,z for Real,
  R for real non empty RelStr,
  a,b for Element of R;

theorem Th10:
  for I being non empty set for J being RelStr-yielding non-Empty
reflexive-yielding ManySortedSet of I st for i being Element of I holds J.i is
  Semilattice holds product J is with_infima
proof
  let I being non empty set;
  let J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I
  such that
A1: for i being Element of I holds J.i is Semilattice;
  set IT = product J;
  for x,y being Element of IT ex z being Element of IT st x >= z & y >= z
  & for z9 being Element of IT st x >= z9 & y >= z9 holds z >= z9
  proof
    let x,y being Element of IT;
    deffunc U(Element of I) = x.$1 "/\" y.$1;
    consider z be ManySortedSet of I such that
A2: for i be Element of I holds z.i = U(i) from PBOOLE:sch 5;
A3: for i being Element of I holds z.i is Element of J.i
    proof
      let i being Element of I;
      z.i = x.i "/\" y.i by A2;
      hence thesis;
    end;
    dom z = I by PARTFUN1:def 2;
    then reconsider z as Element of product J by A3,WAYBEL_3:27;
    take z;
    for i being Element of I holds x.i >= z.i & y.i >= z.i
    proof
      let i being Element of I;
      J.i is antisymmetric with_infima RelStr & z.i = x.i "/\" y.i by A1,A2;
      hence thesis by YELLOW_0:23;
    end;
    hence x >= z & y >= z by WAYBEL_3:28;
    let z9 be Element of IT;
    assume that
A4: x >= z9 and
A5: y >= z9;
    for i being Element of I holds z.i >= z9.i
    proof
      let i being Element of I;
A6:   z9.i <= y.i & z.i = x.i "/\" y.i by A2,A5,WAYBEL_3:28;
      J.i is antisymmetric with_infima RelStr & x.i >= z9.i by A1,A4,
WAYBEL_3:28;
      hence thesis by A6,YELLOW_0:23;
    end;
    hence thesis by WAYBEL_3:28;
  end;
  hence thesis;
end;
