
theorem Th7:
  for S being with_infima Poset, a, b being Element of S holds
  lim_inf Net-Str (a,b) = a "/\" b
proof
  let S be with_infima Poset;
  let a, b be Element of S;
  set N = Net-Str (a,b);
  A1: for
 j be Element of N holds {N.i where i is Element of N : i >= j} = {a,b}
  proof
    let j be Element of N;
    thus {N.i where i is Element of N : i >= j} c= {a,b}
    proof
      let x be object;
      assume x in {N.i where i is Element of N : i >= j};
      then consider i1 be Element of N such that
A2:   x = N.i1 and i1 >= j;
      N.i1 = a or N.i1 = b by Th5;
      hence thesis by A2,TARSKI:def 2;
    end;
    thus {a,b} c= {N.i where i is Element of N : i >= j}
    proof
      let x be object;
      assume
A3:   x in {a,b};
      reconsider J = j as Element of NAT by Def3;
      defpred C[Element of NAT] means ex k be Element of NAT st $1 = 2*k;
      per cases by A3,TARSKI:def 2;
      suppose
A4:     x = a;
        now per cases;
          suppose
A5:         C[J];
A6:         N.j = (a,b),....j by Def3
              .= a by A5,Def1;
            j <= j;
            hence thesis by A4,A6;
          end;
          suppose
A7:         not C[J];
A8:         N.j = (a,b),....j by Def3
              .= b by A7,Def1;
            reconsider k = J + 1 as Element of N by Def3;
A9:         N.k = a by A8,Th6;
            J + 1 >= J by NAT_1:11;
            then k >= j by Def3;
            hence thesis by A4,A9;
          end;
        end;
        hence thesis;
      end;
      suppose
A10:    x = b;
        now per cases;
          suppose
A11:        not C[J];
A12:        N.j = (a,b),....j by Def3
              .= b by A11,Def1;
            j <= j;
            hence thesis by A10,A12;
          end;
          suppose
A13:        C[J];
A14:        N.j = (a,b),....j by Def3
              .= a by A13,Def1;
            reconsider k = J + 1 as Element of N by Def3;
A15:        N.k = b by A14,Th6;
            J + 1 >= J by NAT_1:11;
            then k >= j by Def3;
            hence thesis by A10,A15;
          end;
        end;
        hence thesis;
      end;
    end;
  end;
  defpred P[Element of N,Element of N] means $1 >= $2;
  deffunc F(Element of N) = {N.i1 where i1 is Element of N : P[i1,$1]};
  defpred R[set] means not contradiction;
  deffunc G(Element of N) = {a, b};
  deffunc Q1(Element of N) = "/\"(F($1), S);
  deffunc Q2(Element of N) = "/\"(G($1), S);
  deffunc F(set) = a "/\" b;
A16: for jj be Element of N holds Q1(jj) = F(jj)
  proof
    let jj be Element of N;
    Q1(jj) = Q2(jj) by A1
      .= a "/\" b by YELLOW_0:40;
    hence thesis;
  end;
A17: {Q1(j3) where j3 is Element of N : R[j3]} =
  {F(j4) where j4 is Element of N : R[j4]} from FRAENKEL:sch 5(A16);
A18: {a "/\" b where j4 is Element of N : R[j4]} c= {a "/\" b}
  proof
    let x be object;
    assume x in {a "/\" b where j4 is Element of N : R[j4]};
    then ex q be Element of N st ( x = a "/\" b)&( R[q]);
    hence thesis by TARSKI:def 1;
  end;
  {a "/\" b} c= {a "/\" b where j4 is Element of N : R[j4]}
  proof
    let x be object;
    assume x in {a "/\" b};
    then x = a "/\" b by TARSKI:def 1;
    hence thesis;
  end;
  then {a "/\" b where j4 is Element of N : R[j4]} = {a "/\" b} by A18;
  hence thesis by A17,YELLOW_0:39;
end;
