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*(n+1)) & s.0=0) implies for n st n>=1
  holds Partial_Sums(s).n>(n*(n+1))/2
proof
  defpred X[Nat] means Partial_Sums(s).$1>($1*($1+1))/2;
  assume
A1: for n st n>=1 holds s.n=sqrt(n*(n+1)) & 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>(n*(n+1))/2;
    n+1>=1+1 by A3,XREAL_1:7;
    then n+1>=1 by XXREAL_0:2;
    then
A5: s.(n+1)=sqrt((n+1)*(n+1+1)) by A1;
    n+2>n+1 by XREAL_1:8;
    then (n+2)*(n+1)>(n+1)^2 by XREAL_1:68;
    then sqrt((n+2)*(n+1))>sqrt((n+1)^2) by SQUARE_1:27;
    then
A6: sqrt((n+2)*(n+1))>n+1 by SQUARE_1:22;
    Partial_Sums(s).n+(sqrt((n+1)*(n+2))-((n+1)*(n+2))/2)>(n*(n+1))/2 +(
    sqrt((n+1)*(n+2))-((n+1)*(n+2))/2) by A4,XREAL_1:8;
    then
    Partial_Sums(s).n+sqrt((n+1)*(n+2))-((n+1)*(n+2))/2> sqrt((n+1)*(n+2)
    )-(n+1);
    then Partial_Sums(s).n+sqrt((n+1)*(n+2))-((n+1)*(n+2))/2>0 by A6,XREAL_1:50
;
    then
A7: Partial_Sums(s).n+sqrt((n+1)*(n+2))-((n+1)*(n+2))/2+((n+1)*(n+2))/2>0
    +((n+1)*(n+2))/2 by XREAL_1:8;
    thus thesis by A5,A7,SERIES_1:def 1;
  end;
A8: Partial_Sums(s).1 = Partial_Sums(s).(0+1)
    .=Partial_Sums(s).0+s.1 by SERIES_1:def 1
    .=s.0+s.1 by SERIES_1:def 1
    .=0+s.1 by A1
    .=sqrt(1*(1+1)) by A1
    .=sqrt 2;
  then (Partial_Sums(s).1)^2 = 2 by SQUARE_1:def 2;
  then sqrt((Partial_Sums(s).1)^2)>sqrt(((1*(1+1))/2)^2) by SQUARE_1:27;
  then
A9: sqrt((Partial_Sums(s).1)^2)>((1*(1+1))/2);
  Partial_Sums(s).1>0 by A8,SQUARE_1:25;
  then
A10: X[1] by A9,SQUARE_1:22;
  for n be Nat st n>=1 holds X[n] from NAT_1:sch 8(A10,A2);
  hence thesis;
end;
