
theorem
  for L being antisymmetric with_infima RelStr, A being lower Subset of
  L for B, C being Subset of L holds (A \/ B) "/\" (A \/ C) c= A \/ (B "/\" C)
proof
  let L be antisymmetric with_infima RelStr, A be lower Subset of L, B, C be
  Subset of L;
  let q be object;
  assume q in (A \/ B) "/\" (A \/ C);
  then consider x, y being Element of L such that
A1: q = x "/\" y and
A2: x in A \/ B & y in A \/ C;
A3: x "/\" y <= x by YELLOW_0:23;
A4: x "/\" y <= y by YELLOW_0:23;
  per cases by A2,XBOOLE_0:def 3;
  suppose
    x in A & y in A;
    then q in A by A1,A3,WAYBEL_0:def 19;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
    x in A & y in C;
    then q in A by A1,A3,WAYBEL_0:def 19;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
    x in B & y in A;
    then q in A by A1,A4,WAYBEL_0:def 19;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
    x in B & y in C;
    then x "/\" y in B "/\" C;
    hence thesis by A1,XBOOLE_0:def 3;
  end;
end;
