reserve a,b,c,d for positive Real,
  m,u,w,x,y,z for Real,
  n,k for Nat,
  s,s1 for Real_Sequence;

theorem
  (for n st n>=1 holds s.n=1+1/(2*n+1) & s.0=1) implies for n st n>=1
  holds Partial_Product(s).n>(1/2)*sqrt(2*n+3)
proof
  defpred X[Nat] means Partial_Product(s).$1>(1/2)*sqrt(2*$1+3);
A1: (1/2)*sqrt(2*1+3)=sqrt(5)/2;
  assume
A2: for n st n>=1 holds s.n=1+1/(2*n+1) & s.0=1;
A3: for n be Nat st n>=1 & X[n] holds X[n+1]
  proof
    let n be Nat;
    assume that
A4: n>=1 and
A5: Partial_Product(s).n>(1/2)*sqrt(2*n+3);
    n+1>=1+1 by A4,XREAL_1:7;
    then
A6: n+1>=1 by XXREAL_0:2;
A7: sqrt(2*n+3)>0 by SQUARE_1:25;
    1/(2*n+3)+(2*n+5)>0+(2*n+5) by XREAL_1:8;
    then 1/(2*n+3)+2+(2*n+3)>2*n+5;
    then 1^2/(sqrt(2*n+3))^2+2+(2*n+3)>2*n+5 by SQUARE_1:def 2;
    then (1/(sqrt(2*n+3)))^2+2+(2*n+3)>2*n+5 by XCMPLX_1:76;
    then (1/sqrt(2*n+3))^2+2*1+(sqrt(2*n+3))^2>2*n+5 by SQUARE_1:def 2;
    then
    (1/sqrt(2*n+3))^2+2*(sqrt(2*n+3)*(1/sqrt(2*n+3)))+(sqrt(2*n+3))^2>2*n
    +5 by A7,XCMPLX_1:106;
    then sqrt((1/sqrt(2*n+3)+sqrt(2*n+3))^2)>sqrt(2*n+5) by SQUARE_1:27;
    then sqrt(2*n+3)+1/sqrt(2*n+3)>sqrt(2*n+5) by A7,SQUARE_1:22;
    then
A8: (1/2)*(sqrt(2*n+3)+1/sqrt(2*n+3))>(1/2)*sqrt(2*n+5) by XREAL_1:68;
    Partial_Product(s).n*(1+1/(2*n+3))>((1/2)*sqrt(2*n+3))*(1+1/(2*n+3))
    by A5,XREAL_1:68;
    then
    Partial_Product(s).n*(1+1/(2*n+3))>((1/2)*sqrt(2*n+3))*1+ ((1/2)*sqrt
    (2*n+3))*(1/(2*n+3));
    then
    Partial_Product(s).n*(1+1/(2*n+3))>(1/2)*sqrt(2*n+3)+ ((1/2)*sqrt(2*n
    +3))/(2*n+3) by XCMPLX_1:74;
    then
    Partial_Product(s).n*(1+1/(2*n+3))>(1/2)*sqrt(2*n+3)+ (1/2)*(sqrt(2*n
    +3)/(2*n+3)) by XCMPLX_1:74;
    then
    Partial_Product(s).n*(1+1/(2*n+3))>(1/2)*sqrt(2*n+3)+ (1/2)*(sqrt(2*n
    +3)/sqrt((2*n+3)^2)) by SQUARE_1:22;
    then
    Partial_Product(s).n*(1+1/(2*n+3))>(1/2)*sqrt(2*n+3)+ (1/2)*((sqrt(2*
    n+3)*1)/(sqrt(2*n+3)*sqrt(2*n+3))) by SQUARE_1:29;
    then
    Partial_Product(s).n*(1+1/(2*n+3))>(1/2)*sqrt(2*n+3)+ (1/2)*((sqrt(2*
    n+3)/sqrt(2*n+3))*(1/sqrt(2*n+3))) by XCMPLX_1:76;
    then Partial_Product(s).n*(1+1/(2*n+3))>(1/2)*sqrt(2*n+3)+ (1/2)*(1*(1/
    sqrt(2*n+3))) by A7,XCMPLX_1:60;
    then Partial_Product(s).n*(1+1/(2*(n+1)+1))>(1/2)*sqrt(2*n+5) by A8,
XXREAL_0:2;
    then Partial_Product(s).n*s.(n+1)>(1/2)*sqrt(2*n+5) by A2,A6;
    hence thesis by SERIES_3:def 1;
  end;
  sqrt(16/9)>sqrt(5/4) by SQUARE_1:27;
  then sqrt(4^2)/sqrt(3^2)>sqrt(5/4) by SQUARE_1:30;
  then 4/sqrt(3^2)>sqrt(5/4) by SQUARE_1:22;
  then 4/3>sqrt(5/4) by SQUARE_1:22;
  then
A9: 4/3>sqrt(5)/sqrt(2^2) by SQUARE_1:30;
  Partial_Product(s).(1+0) =Partial_Product(s).0 * s.(1+0) by SERIES_3:def 1
    .=s.0 * s.1 by SERIES_3:def 1
    .=1*s.1 by A2
    .=1+1/(2*1+1) by A2
    .=4/3;
  then
A10: X[1] by A1,A9,SQUARE_1:22;
  for n be Nat st n>=1 holds X[n] from NAT_1:sch 8(A10,A3);
  hence thesis;
end;
