
theorem Th8:
  for S, T being with_infima Poset, a, b being Element of S,
  f being Function of S, T holds lim_inf (f*Net-Str (a,b)) = f.a "/\" f.b
proof
  let S, T be with_infima Poset;
  let a, b be Element of S;
  let f be Function of S, T;
  set N = Net-Str (a,b);
  set fN = f * N;
A1: the RelStr of fN = the RelStr of N by WAYBEL_9:def 8;
A2: for j being Element of fN holds
  {fN.i where i is Element of fN : i >= j} = {f.a, f.b}
  proof
    let j be Element of fN;
    reconsider jj = j as Element of N by A1;
    thus {fN.i where i is Element of fN : i >= j} c= {f.a,f.b}
    proof
      let x be object;
      assume x in {fN.i where i is Element of fN : i >= j};
      then consider i1 be Element of fN such that
A3:   x = fN.i1 and i1 >= j;
      reconsider I1 = i1 as Element of N by A1;
      i1 in the carrier of N by A1;
      then
A4:   i1 in dom the mapping of N by FUNCT_2:def 1;
      fN.i1 = (f * the mapping of N).i1 by WAYBEL_9:def 8
        .= f.(N.I1) by A4,FUNCT_1:13;
      then fN.i1 = f.a or fN.i1 = f.b by Th5;
      hence thesis by A3,TARSKI:def 2;
    end;
    thus {f.a,f.b} c= {fN.i where i is Element of fN : i >= j}
    proof
      let x be object;
      assume
A5:   x in {f.a,f.b};
A6:   j in the carrier of N by A1;
      reconsider J = j as Element of NAT by A1,Def3;
A7:   j in dom the mapping of N by A6,FUNCT_2:def 1;
      defpred C[Element of NAT] means ex k be Element of NAT st $1 = 2*k;
      per cases by A5,TARSKI:def 2;
      suppose
A8:     x = f.a;
        reconsider jj = j as Element of N by A1;
        now per cases;
          suppose
A9:         C[J];
A10:        fN.j = (f * the mapping of N).j by WAYBEL_9:def 8
              .= f.((the mapping of N).j) by A7,FUNCT_1:13
              .= f.((a,b),....j) by Def3
              .= f.a by A9,Def1;
            j <= j;
            hence thesis by A8,A10;
          end;
          suppose
A11:        not C[J];
A12:        N.jj = (a,b),....jj by Def3
              .= b by A11,Def1;
            reconsider k = J + 1 as Element of fN by A1,Def3;
            reconsider kk = k as Element of N by A1;
            kk in the carrier of N;
            then
A13:        kk in dom the mapping of N by FUNCT_2:def 1;
A14:        fN.k = (f * the mapping of N).k by WAYBEL_9:def 8
              .= f.(N.kk) by A13,FUNCT_1:13
              .= f.a by A12,Th6;
            J + 1 >= J by NAT_1:11;
            then kk >= jj by Def3;
            then [jj,kk] in the InternalRel of N by ORDERS_2:def 5;
            then k >= j by A1,ORDERS_2:def 5;
            hence thesis by A8,A14;
          end;
        end;
        hence thesis;
      end;
      suppose
A15:    x = f.b;
        now per cases;
          suppose
A16:        not C[J];
A17:        fN.j = (f * the mapping of N).j by WAYBEL_9:def 8
              .= f.((the mapping of N).j) by A7,FUNCT_1:13
              .= f.((a,b),....j) by Def3
              .= f.b by A16,Def1;
            j <= j;
            hence thesis by A15,A17;
          end;
          suppose
A18:        C[J];
A19:        N.jj = (a,b),....j by Def3
              .= a by A18,Def1;
            reconsider k = J + 1 as Element of fN by A1,Def3;
            reconsider kk = k as Element of N by Def3;
            kk in the carrier of N;
            then
A20:        kk in dom the mapping of N by FUNCT_2:def 1;
A21:        fN.k = (f * the mapping of N).k by WAYBEL_9:def 8
              .= f.(N.kk) by A20,FUNCT_1:13
              .= f.b by A19,Th6;
            J + 1 >= J by NAT_1:11;
            then kk >= jj by Def3;
            then [jj,kk] in the InternalRel of N by ORDERS_2:def 5;
            then k >= j by A1,ORDERS_2:def 5;
            hence thesis by A15,A21;
          end;
        end;
        hence thesis;
      end;
    end;
  end;
  defpred P[Element of fN,Element of fN] means $1 >= $2;
  deffunc F(Element of fN) = {fN.i1 where i1 is Element of fN : P[i1,$1]};
  defpred R[set] means not contradiction;
  deffunc G(Element of fN) = {f.a, f.b};
  deffunc Q1(Element of fN) = "/\"(F($1), T);
  deffunc Q2(Element of fN) = "/\"(G($1), T);
  deffunc F(set) = f.a "/\" f.b;
A22: for jj be Element of fN holds Q1(jj) = F(jj)
  proof
    let jj be Element of fN;
    Q1(jj) = Q2(jj) by A2
      .= f.a "/\" f.b by YELLOW_0:40;
    hence thesis;
  end;
A23: {Q1(j3) where j3 is Element of fN : R[j3]} =
  {F(j4) where j4 is Element of fN : R[j4]} from FRAENKEL:sch 5(A22);
  A24: {f.a "/\" f.b where j4 is Element of fN : R[j4]} c= {f.a "/\" f.b}
  proof
    let x be object;
    assume x in {f.a "/\" f.b where j4 is Element of fN : R[j4]};
    then ex q be Element of fN st ( x = f.a "/\" f.b)&( R[q]);
    hence thesis by TARSKI:def 1;
  end;
  {f.a "/\" f.b} c= {f.a "/\" f.b where j4 is Element of fN : R[j4]}
  proof
    let x be object;
    assume x in {f.a "/\" f.b};
    then x = f.a "/\" f.b by TARSKI:def 1;
    hence thesis;
  end;
  then {f.a "/\" f.b where j4 is Element of fN : R[j4]} = {f.a "/\" f.b} by A24
;
  hence thesis by A23,YELLOW_0:39;
end;
