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=(2*n+1)/(2*n+2)) implies (Partial_Product s).n<=1/
  sqrt(3*n+4)
proof
  defpred X[Nat] means (Partial_Product s).$1<=1/sqrt(3*$1+4);
  assume
A1: for n holds s.n=(2*n+1)/(2*n+2);
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume (Partial_Product s).n<=1/sqrt(3*n+4);
    then
    (Partial_Product s).n*((2*n+3)/(2*n+4))<=1/sqrt(3*n+4)*((2*n+3)/(2*n+
    4 )) by XREAL_1:64;
    then
A3: (Partial_Product s).n*((2*n+3)/(2*n+4))<=(1*(2*n+3))/(sqrt(3*n+4)* (2*
    n+ 4)) by XCMPLX_1:76;
    111*n<=112*n by XREAL_1:64;
    then 111*n+63<=112*n+64 by XREAL_1:7;
    then (12*n|^3+64*n^2)+(111*n+63)<=(12*n|^3+64*n^2)+(112*n+64) by XREAL_1:6;
    then
    12*n|^(2+1)+36*(n*n)+27*n+28*n^2+84*n+63<=12*n|^(2+1)+48*n^2+48*n+16*
    n ^2 +64*n+64;
    then
    12*((n|^2)*(n|^1))+36*(n*n)+27*n+28*n^2+84*n+63<=12*n|^(2+1)+48*(n*n)
    + 48*n+16*n^2+64*n+64 by NEWTON:8;
    then
    4*(n|^2)*3*(n|^1)+36*n*n+27*n+28*n^2+84*n+63<=12*n|^(2+1)+48*n*n+ 48*
    n+16*n^2+64*n+64;
    then
    4*(n|^2)*3*n+36*n*n+27*n+28*n^2+84*n+63<=12*n|^(2+1)+48*n*n+ 48*n+16*
    n^2+64*n+64;
    then
    4*(n^2)*3*n+36*n*n+27*n+28*n^2+84*n+63<=12*n|^(2+1)+48*n*n+ 48*n+16*n
    ^2+64*n+64 by Lm1;
    then 4*(n^2)*3*n+36*n*n+27*n+28*n^2+84*n+63<=12*(n|^2*n|^1)+48*n*n+ 48*n+
    16*n^2+64*n+64 by NEWTON:8;
    then
    4*(n^2)*3*n+36*n*n+27*n+28*n^2+84*n+63<=12*(n^2*n|^1)+48*n*n+ 48*n+16
    *n^2+64*n+64 by Lm1;
    then 4*(n^2)*3*n+36*n*n+27*n+28*n^2+84*n+63<=12*(n^2*n)+48*n*n+ 48*n+16*n
    ^2+64*n+64;
    then sqrt(((2*n+3)^2)*((3*n)+7))<=sqrt(((2*n)+4)^2 *((3*n)+4)) by
SQUARE_1:26;
    then sqrt((2*n+3)^2)*sqrt((3*n)+7)<=sqrt(((2*n)+4)^2 *((3*n)+4)) by
SQUARE_1:29;
    then sqrt((2*n+3)^2)*sqrt((3*n)+7)<=sqrt((2*n+4)^2)*sqrt((3*n+4)) by
SQUARE_1:29;
    then (2*n+3)*sqrt((3*n)+7)<=sqrt((2*n+4)^2)*sqrt((3*n+4)) by SQUARE_1:22;
    then
A4: (2*n+3)*sqrt((3*n)+7)<=((2*n+4)*sqrt((3*n+4)))*1 by SQUARE_1:22;
    sqrt(3*n+4)>0 & sqrt(3*n+7)>0 by SQUARE_1:25;
    then 1*(2*n+3)/((2*n+4)*sqrt((3*n+4)))<=1/sqrt((3*n)+7) by A4,XREAL_1:102;
    then
    (Partial_Product s).n*((2*(n+1)+1)/(2*(n+1)+2))<=1/sqrt(3*(n+1)+4) by A3,
XXREAL_0:2;
    then (Partial_Product s).n*s.(n+1)<=1/sqrt(3*(n+1)+4) by A1;
    hence thesis by Def1;
  end;
  (Partial_Product s).0 =s.0 by Def1
    .=(2*0+1)/(2*0+2) by A1
    .=1/sqrt(3*0+4) by SQUARE_1:20;
  then
A5: X[0];
  for n holds X[n] from NAT_1:sch 2(A5,A2);
  hence thesis;
end;
