
theorem Th7:
  for L being antisymmetric reflexive with_infima RelStr, x being
  Element of L holds (x"/\")"{x} = uparrow x
proof
  let L be antisymmetric reflexive with_infima RelStr, x be Element of L;
  thus (x"/\")"{x} c= uparrow x
  proof
    let q be object;
    assume
A1: q in (x"/\")"{x};
    then reconsider q1 = q as Element of L;
A2: (x"/\").q1 in {x} by A1,FUNCT_1:def 7;
    x "/\" q1 = (x"/\").q1 by WAYBEL_1:def 18
      .= x by A2,TARSKI:def 1;
    then x <= q1 by YELLOW_0:25;
    hence thesis by WAYBEL_0:18;
  end;
  let q be object;
  assume
A3: q in uparrow x;
  then reconsider q1 = q as Element of L;
A4: x <= q1 by A3,WAYBEL_0:18;
  (x"/\").q1 = x "/\" q1 by WAYBEL_1:def 18
    .= x by A4,YELLOW_0:25;
  then dom (x"/\") = the carrier of L & (x"/\").q1 in {x} by FUNCT_2:def 1
,TARSKI:def 1;
  hence thesis by FUNCT_1:def 7;
end;
