theorem
  (for n being Nat holds S3.n = S1.n /\ S2.n)
implies lim_inf S3 = lim_inf S1 /\
  lim_inf S2
proof
A1: (for n being Nat holds A3.n = B.n /\ A2.n)
implies Union inferior_setsequence(B)
  /\ Union inferior_setsequence(A2) = Union inferior_setsequence(A3)
  proof
A2: lim_inf B = Union inferior_setsequence(B) & lim_inf A2 = Union
    inferior_setsequence(A2);
A3: lim_inf A3 = Union inferior_setsequence(A3);
    assume for n being Nat holds A3.n = B.n /\ A2.n;
    hence thesis by A2,A3,KURATO_0:10;
  end;
  assume for n being Nat holds S3.n = S1.n /\ S2.n;
  hence thesis by A1;
end;
