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=sqrt(n) & s.0=0) implies for n st n>=1 holds
  Partial_Sums(s).n>(2/3)*n*sqrt(n)
proof
  defpred X[Nat] means Partial_Sums(s).$1>(2/3)*$1*sqrt($1);
  assume
A1: for n st n>=1 holds s.n=sqrt(n) & s.0=0;
A2: for n be Nat st n>=1 & X[n] holds X[n+1]
  proof
    let n be Nat;
    assume that
A3: n>=1 and
A4: Partial_Sums(s).n>(2/3)*n*sqrt(n);
    2*n>=2*1 by A3,XREAL_1:64;
    then 2*n>1 by XXREAL_0:2;
    then
A5: 2*n-1>0 by XREAL_1:50;
    3*n>=3*1 by A3,XREAL_1:64;
    then 3*n>1 by XXREAL_0:2;
    then 1-3*n<0 by XREAL_1:49;
    then 1+(n-4*n)+(4*n^2-4*n^2)+4*n|^3<0+4*n|^3 by XREAL_1:8;
    then (4*(n|^(2+1))-4*n^2+n)+(4*n^2-4*n+1)<4*n|^3;
    then (4*(n|^2*n)-4*(n*n)+n)+(4*n^2-4*n+1)<4*n|^3 by NEWTON:6;
    then (4*n|^2-4*n+1)*n+(4*n^2-4*n+1)*1<4*n|^3;
    then (4*n^2-4*n+1)*n+(4*n^2-4*n+1)*1<4*n|^3 by NEWTON:81;
    then (4*n^2-4*n+1)*(n+1)<4*n|^(2+1);
    then (4*n^2-4*n+1)*(n+1)<4*(n|^2*n) by NEWTON:6;
    then (4*n^2-4*n+1)*(n+1)<4*n|^2*n;
    then (2*n-1)^2*(n+1)<4*n^2*n by NEWTON:81;
    then (n+1)/n<(2*n)^2/(2*n-1)^2 by A3,A5,XREAL_1:106;
    then (n+1)/n<((2*n)/(2*n-1))^2 by XCMPLX_1:76;
    then sqrt((n+1)/n)<sqrt(((2*n)/(2*n-1))^2) by SQUARE_1:27;
    then sqrt((n+1)/n)<(2*n)/(2*n-1) by A5,SQUARE_1:22;
    then
A6: sqrt(n+1)/sqrt(n)<(2*n)/(2*n-1) by SQUARE_1:30;
    sqrt(n)>0 by A3,SQUARE_1:25;
    then (1/3)*(sqrt(n+1)*(2*n-1))<(1/3)*((2*n)*sqrt(n)) by A5,A6,XREAL_1:68
,102;
    then Partial_Sums(s).n>((2/3)*(n+1))*sqrt(n+1)-sqrt(n+1) by A4,XXREAL_0:2;
    then
A7: Partial_Sums(s).n+sqrt(n+1)>((2/3)*(n+1))*sqrt(n+1)-sqrt(n+1)+ sqrt(n
    +1 ) by XREAL_1:8;
    n+1>=1+1 by A3,XREAL_1:7;
    then n+1>=1 by XXREAL_0:2;
    then Partial_Sums(s).n+s.(n+1)>((2/3)*(n+1))*sqrt(n+1) by A1,A7;
    hence thesis by SERIES_1:def 1;
  end;
  Partial_Sums(s).(1+0) =Partial_Sums(s).0 + s.(1+0) by SERIES_1:def 1
    .=s.0 + s.1 by SERIES_1:def 1
    .=0+s.1 by A1
    .=1 by A1,SQUARE_1:18;
  then
A8: X[1];
  for n be Nat st n>=1 holds X[n] from NAT_1:sch 8(A8,A2);
  hence thesis;
end;
