reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  (for n holds s.n>0 & s.n>=s.(n-1)) implies (Partial_Sums s).n>=(n+1)*(
  (n+1)-root((Partial_Product s).n))
proof
  defpred X[Nat] means (Partial_Sums s).$1>= ($1+1)*(($1+1)-root((
  Partial_Product s).$1));
A1: (Partial_Sums s).0=s.0 by SERIES_1:def 1;
  assume
A2: for n holds s.n>0 & s.n>=s.(n-1);
A3: for n st X[n] holds X[n+1]
  proof
    let n;
    set u=(Partial_Sums s).n;
    set v=s.(n+1);
    set w=((u/(n+1))+(v-(u/(n+1)))/(n+2))|^(n+2);
    set h=((n+1)-root((Partial_Product s).n));
    set j=u/(n+1);
A4: v>0 by A2;
A5: n+1>=0+1 by XREAL_1:6;
    then (n+1)+1>=1+1 by XREAL_1:6;
    then
A6: n+2>=1 by XXREAL_0:2;
    assume (Partial_Sums s).n>=(n+1)*((n+1)-root((Partial_Product s).n));
    then u/(n+1)>=((n+1)* ((n+1)-root((Partial_Product s).n)))/(n+1) by
XREAL_1:72;
    then ((n+1)-root((Partial_Product s).n))<=u/(n+1) by XCMPLX_1:89;
    then
A7: h|^(n+1)<=j|^(n+1) by A2,Lm11,PREPOWER:9;
A8: (Partial_Product(s)).n>0 by A2,Th43;
    then h=((n+1) -Root((Partial_Product s).n)) by A5,POWER:def 1;
    then (Partial_Product s).n<=j|^(n+1) by A7,A8,A5,PREPOWER:19;
    then ((Partial_Product s).n)*v<=(j|^(n+1))*v by A4,XREAL_1:64;
    then
A9: ((Partial_Product s).(n+1))<=(j|^(n+1))*v by Def1;
A10: (Partial_Product s).(n+1)>=0 by A2,Th43;
    u>0 & (v-(u/(n+1)))/(n+2)>=0 by A2,Lm12,Th33;
    then
    w>=(u/(n+1))|^((n+1)+1)+(n+2)*((u/(n+1))|^(n+1))* ((v-(u/(n+1)))/(n+2
    )) by Th31;
    then
    w>=(u/(n+1))|^((n+1))*(u/(n+1))+((u/(n+1))|^(n+1))* ((v-(u/(n+1)))/(n
    +2)*(n+2)) by NEWTON:6;
    then
A11: w>=(u/(n+1))|^((n+1))*(u/(n+1))+((u/(n+1))|^(n+1))* (v-(u/(n+1))) by
XCMPLX_1:87;
A12: (Partial_Sums s).(n+1)>0 by A2,Th33;
    ((Partial_Sums s).(n+1))/(n+2) =(1*(Partial_Sums s).n+s.(n+1))/(n+2)
    by SERIES_1:def 1
      .=((n+1)/(n+1)*(Partial_Sums s).n+s.(n+1))/(n+2) by XCMPLX_1:60
      .=((((n+2)-1)*(Partial_Sums s).n)/(n+1)+s.(n+1))/(n+2);
    then ((Partial_Sums s).(n+1))/(n+2) =(((n+2)*u-u)/(n+1)+v)/(n+2)
      .=((((n+2)*u)/(n+1)-u/(n+1))+v)/(n+2)
      .=(((n+2)*u)/(n+1)-u/(n+1))/(n+2)+v/(n+2)
      .=((n+2)*u)/(n+1)/(n+2)-(u/(n+1))/(n+2)+v/(n+2)
      .=((n+2)*u)/((n+1)*(n+2))-(u/(n+1))/(n+2)+v/(n+2) by XCMPLX_1:78
      .=u/(n+1)-(u/(n+1))/(n+2)+v/(n+2) by XCMPLX_1:91
      .=u/(n+1)+(v/(n+2)-(u/(n+1))/(n+2))
      .=u/(n+1)+(v-(u/(n+1)))/(n+2);
    then (((Partial_Sums s).(n+1))/(n+2))|^(n+2)>= (Partial_Product s).(n+1)
    by A11,A9,XXREAL_0:2;
    then (n+2) -Root((((Partial_Sums s).(n+1))/(n+2))|^(n+2))>= (n+2) -Root((
    Partial_Product s).(n+1)) by A6,A10,PREPOWER:27;
    then
    (((Partial_Sums s).(n+1))/(n+2))>= (n+2) -Root((Partial_Product s).(n
    +1)) by A6,A12,PREPOWER:19;
    then
    (((Partial_Sums s).(n+1))/(n+2))>= (n+2)-root((Partial_Product s).(n+
    1)) by A6,A10,POWER:def 1;
    then ((Partial_Sums s).(n+1))/(n+2)*(n+2)>= (n+2)*((n+2)-root((
    Partial_Product s).(n+1))) by XREAL_1:64;
    hence thesis by XCMPLX_1:87;
  end;
  (0+1)*((0+1)-root((Partial_Product s).0)) =((0+1)-root(s.0)) by Def1
    .=(Partial_Sums s).0 by A1,POWER:9;
  then
A13: X[0];
  for n holds X[n] from NAT_1:sch 2(A13,A3);
  hence thesis;
end;
