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 :: Bernoulli Inequality(1)
  (for n holds s1.n=1+s.n & s.n>-1 & s.n<0) implies for n holds 1+
  Partial_Sums(s).n<=(Partial_Product s1).n
proof
  defpred X[Nat] means 1+Partial_Sums(s).$1<=(Partial_Product s1).
  $1;
  assume
A1: for n holds s1.n=1+s.n & s.n>-1 & s.n<0;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
A3: ((Partial_Product(s1)).n)*(1+s.(n+1))=((Partial_Product s1).n)*s1.(n+
    1) by A1;
    s.(n+1)>-1 by A1;
    then
A4: s.(n+1)+1>-1+1 by XREAL_1:8;
    assume 1+Partial_Sums(s).n<=(Partial_Product s1).n;
    then
    (1+Partial_Sums(s).n)*(1+s.(n+1))<=((Partial_Product s1).n)*(1+s.( n+1
    )) by A4,XREAL_1:64;
    then
A5: (1+Partial_Sums(s).n)*(1+s.(n+1))<=(Partial_Product s1).(n+1) by A3,Def1;
    (Partial_Sums(s).n)*(s.(n+1))>=0 by A1,Lm8;
    then
A6: (Partial_Sums(s).n)*(s.(n+1))+(1+s.(n+1)+Partial_Sums(s).n)>= 0+(1+s.
    (n+1)+Partial_Sums(s).n) by XREAL_1:6;
    1+Partial_Sums(s).(n+1)=1+(Partial_Sums(s).n+s.(n+1)) by SERIES_1:def 1;
    hence thesis by A5,A6,XXREAL_0:2;
  end;
  let n;
  Partial_Sums(s).0=s.0 & (Partial_Product s1).0=s1.0 by Def1,SERIES_1:def 1;
  then
A7: X[0] by A1;
  for n holds X[n] from NAT_1:sch 2(A7,A2);
  hence thesis;
end;
