reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem
  for K, L, e being Real st 0 < K & K < 1 & 0 < e
  ex n being Nat st |. L*(K to_power n) .| < e
  proof
    let K, L, e be Real such that
A1: 0 < K and
A2: K < 1 and
A3: 0 < e;
    deffunc P(Nat) = K to_power ($1+1);
    consider rseq be Real_Sequence such that
A4: for n be Nat holds rseq.n=P(n) from SEQ_1:sch 1;
A5: L(#)rseq is convergent by A1,A2,A4,SEQ_2:7,SERIES_1:1;
A6: lim (L(#)rseq) = L* lim rseq by A1,A2,A4,SEQ_2:8,SERIES_1:1
    .=L * (0 qua Real) by A1,A2,A4,SERIES_1:1;
    consider n be Nat such that
A7: for m be Nat st n<=m holds
    |.((L(#)rseq).m-(0 qua Real)).| < e by A5,A6,A3,SEQ_2:def 7;
    |.((L(#)rseq).n-(0 qua Real)).| < e by A7;
    then |. L*rseq.n .| < e by SEQ_1:9;
    then |. L*(K to_power (n+1)) .| < e by A4;
    hence thesis;
  end;
