
theorem Th50:
  for L being antisymmetric reflexive with_infima RelStr for A, B
  being lower Subset of L holds A "/\" B = A /\ B
proof
  let L be antisymmetric reflexive with_infima RelStr, A, B be lower Subset of
  L;
  thus A "/\" B c= A /\ B
  proof
    let q be object;
    assume q in A "/\" B;
    then consider x, y being Element of L such that
A1: q = x "/\" y and
A2: x in A and
A3: y in B;
A4: ex z being Element of L st x >= z & y >= z & for c being Element of L
    st x >= c & y >= c holds z >= c by LATTICE3:def 11;
    then x "/\" y <= y by LATTICE3:def 14;
    then
A5: q in B by A1,A3,WAYBEL_0:def 19;
    x "/\" y <= x by A4,LATTICE3:def 14;
    then q in A by A1,A2,WAYBEL_0:def 19;
    hence thesis by A5,XBOOLE_0:def 4;
  end;
  thus thesis by Th49;
end;
