
theorem Th37:
  for a be NAT-valued Real_Sequence, b be non trivial Nat st
  rng a c= b & a is eventually-non-zero holds
  for n be non zero Nat ex p be Integer, q be Nat st q > 1 &
  0 < |. Liouville_constant (a,b) - p/q .| < 1/q|^n
  proof
    let a be NAT-valued Real_Sequence, b be non trivial Nat;
    assume
A1: rng a c= b & a is eventually-non-zero;
    set x = Liouville_constant (a,b);
A2: b >= 1 + 1 by NAT_2:29; then
A3: b > 1 by NAT_1:13;
    set pn = ALiouville_seq (a,b);
    set qn = BLiouville_seq b;
    let n be non zero Nat;
A4: n >= 0 + 1 by NAT_1:13;
    reconsider p = pn.n as Integer;
    reconsider q = qn.n as Nat;
    take p, q;
A5: q = b to_power (n!) by LiuSeq;
      thus q > 1 by Th30,A3;
      set LS = Liouville_seq (a,b);
A6:   LS is summable by Th31,NAT_2:29,A1;
A7:   Sum (Liouville_seq (a,b)^\(n+1)) > 0 by Th36,A1;
      LS.0 = 0 by DefLio; then
A8:   Sum LS = Sum FinSeq (LS,n) + Sum (LS^\(n+1)) by Th23,A6;
A9:   |. x - p/q .|
    = |. Sum Liouville_seq(a,b) - Sum FinSeq(Liouville_seq (a,b),n) .|
      by A3,Th32
   .= Sum (Liouville_seq (a,b)^\(n+1)) by A7,A8,ABSVALUE:def 1;
A10:    Sum (Liouville_seq (a,b)^\(n+1))
    <= Sum ((b-1)(#)((powerfact b)^\(n+1)))  by A1,A2,A3,Th35;
       Sum ((b-1) (#) ((powerfact b)^\(n+1)))
     < 1/((b to_power (n!)) to_power n) by A3,A4,Th27;
    hence thesis by A1,A5,A9,A10,XXREAL_0:2,Th36;
  end;
