
theorem
  for X being set, A, B, C being SetSequence of X
   st for n being Nat holds C.n = A.n /\ B.n
 holds lim_inf C = lim_inf A /\ lim_inf B
proof
  let X be set, A, B, C be SetSequence of X;
  assume
A1: for n being Nat holds C.n = A.n /\ B.n;
  thus lim_inf C c= lim_inf A /\ lim_inf B
  proof
    let x be object;
    assume x in lim_inf C;
    then consider n being Nat such that
A2: for k being Nat holds x in C.(n+k) by Th4;
    for k being Nat holds x in B.(n+k)
    proof
      let k be Nat;
      C.(n+k) = A.(n+k) /\ B.(n+k) & x in C.(n+k) by A1,A2;
      hence thesis by XBOOLE_0:def 4;
    end;
    then
A3: x in lim_inf B by Th4;
    for k being Nat holds x in A.(n+k)
    proof
      let k be Nat;
      C.(n+k) = A.(n+k) /\ B.(n+k) & x in C.(n+k) by A1,A2;
      hence thesis by XBOOLE_0:def 4;
    end;
    then x in lim_inf A by Th4;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  thus lim_inf A /\ lim_inf B c= lim_inf C
  proof
    let x be object;
    assume
A4: x in lim_inf A /\ lim_inf B;
    then x in lim_inf A by XBOOLE_0:def 4;
    then consider n1 being Nat such that
A5: for k being Nat holds x in A.(n1+k) by Th4;
    x in lim_inf B by A4,XBOOLE_0:def 4;
    then consider n2 being Nat such that
A6: for k being Nat holds x in B.(n2+k) by Th4;
    set n = max (n1, n2);
A0: n is Nat by TARSKI:1;
A7: for k being Nat holds x in B.(n+k)
    proof
      let k be Nat;
      consider g being Nat such that
A8:   n = n2 + g by NAT_1:10,XXREAL_0:25;
      reconsider g as Nat;
      x in B.(n2+(g+k)) by A6;
      hence thesis by A8;
    end;
A9: for k being Nat holds x in A.(n+k)
    proof
      let k be Nat;
      consider g being Nat such that
A10:  n = n1 + g by NAT_1:10,XXREAL_0:25;
      reconsider g as Nat;
      x in A.(n1+(g+k)) by A5;
      hence thesis by A10;
    end;
    for k being Nat holds x in C.(n+k)
    proof
      let k be Nat;
      x in A.(n+k) & x in B.(n+k) by A9,A7;
      then x in A.(n+k) /\ B.(n+k) by XBOOLE_0:def 4;
      hence thesis by A1;
    end;
    hence thesis by A0,Th4;
  end;
end;
