reserve n,m,k,k1,k2,i,j for Nat;
reserve x,y,z for object,X,Y,Z for set;
reserve A for Subset of X;
reserve B,A1,A2,A3 for SetSequence of X;
reserve Si for SigmaField of X;
reserve S,S1,S2,S3 for SetSequence of Si;

theorem Th67:
  x in lim_inf S iff ex n being Nat st for k being Nat holds x in S.(n+k)
proof
  x in Union inferior_setsequence(B) iff
   ex n being Nat st for k being Nat holds x in B.(n+k)
  proof
    lim_inf B = Union inferior_setsequence(B);
    hence thesis by KURATO_0:4;
  end;
  hence thesis;
end;
