
theorem Th7:
  for ps being non empty FinSequence of REAL, x being Real st x >=
1 & for i being Element of NAT st i in dom ps holds ps.i > x holds Product(ps)
  > x
proof
  let ps be non empty FinSequence of REAL, x be Real such that
A1: x >= 1 and
A2: for j being Element of NAT st j in dom ps holds ps.j > x;
  consider ps1 being FinSequence, y being object such that
A3: ps = ps1^<*y*> by FINSEQ_1:46;
  <*y*> is FinSequence of REAL by A3,FINSEQ_1:36;
  then rng <*y*> c= REAL by FINSEQ_1:def 4;
  then {y} c= REAL by FINSEQ_1:38;
  then reconsider y2=y as Element of REAL by ZFMISC_1:31;
  defpred P[Nat] means for f being FinSequence of REAL st len f =
$1 & for j being Element of NAT st j in dom f holds f.j > x holds Product(f)*y2
  > x;
A4: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A5: P[k];
    let f be FinSequence of REAL such that
A6: len f = k+1 and
A7: for j being Element of NAT st j in dom f holds f.j > x;
    f <> {} by A6;
    then consider v being FinSequence, w being object such that
A8: f=v^<*w*> by FINSEQ_1:46;
    reconsider f1=v, f2=<*w*> as FinSequence of REAL by A8,FINSEQ_1:36;
    rng f2 c= REAL;
    then {w} c= REAL by FINSEQ_1:38;
    then reconsider m=w as Element of REAL by ZFMISC_1:31;
    k + 1 = len f1 + len f2 by A6,A8,FINSEQ_1:22;
    then
A9: k + 1 = len f1 + 1 by FINSEQ_1:39;
    then
A10: f.(len f) = m by A6,A8,FINSEQ_1:42;
    len f in Seg len f by A6,FINSEQ_1:3;
    then
A11: len f in dom f by FINSEQ_1:def 3;
    then m > 1 by A1,A7,A10,XXREAL_0:2;
    then
A12: x*m > x by A1,XREAL_1:155;
A13: for j being Element of NAT st j in dom f1 holds f1.j > x
    proof
A14:  dom f1 c= dom f by A8,FINSEQ_1:26;
      let j be Element of NAT such that
A15:  j in dom f1;
      f.j = f1.j by A8,A15,FINSEQ_1:def 7;
      hence thesis by A7,A15,A14;
    end;
    Product f = Product f1 * m by A8,RVSUM_1:96;
    then
A16: Product f*y2 = (Product f1 * y2) * m;
    m > x by A7,A10,A11;
    then Product f*y2 > x*m by A1,A5,A9,A13,A16,XREAL_1:68;
    hence thesis by A12,XXREAL_0:2;
  end;
  len ps in Seg len ps by FINSEQ_1:3;
  then
A17: len ps in dom ps by FINSEQ_1:def 3;
  reconsider q=ps1 as FinSequence of REAL by A3,FINSEQ_1:36;
A18: for j being Element of NAT st j in dom q holds q.j > x
  proof
A19: dom q c= dom ps by A3,FINSEQ_1:26;
    let j be Element of NAT such that
A20: j in dom q;
    ps.j = q.j by A3,A20,FINSEQ_1:def 7;
    hence thesis by A2,A20,A19;
  end;
A21: len q = len q;
  len ps = len ps1 + len <*y*> by A3,FINSEQ_1:22;
  then len ps = len ps1 + 1 by FINSEQ_1:39;
  then
A22: ps.(len ps) = y2 by A3,FINSEQ_1:42;
A23: P[0]
  proof
    let f be FinSequence of REAL such that
A24: len f = 0 and
    for j being Element of NAT st j in dom f holds f.j > x;
    f = <*>REAL by A24;
    then Product f = 1 by RVSUM_1:94;
    hence thesis by A2,A22,A17;
  end;
  for k being Nat holds P[k] from NAT_1:sch 2(A23,A4);
  then Product(q)*y2 > x by A18,A21;
  hence thesis by A3,RVSUM_1:96;
end;
