reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem Th49:
  Sum powersFS(-k,3,2*k) = k|^3
  proof
    defpred P[Nat] means Sum powersFS(-$1,3,2*$1) = $1|^3;
A1: P[0]
    proof
      set f = powersFS(-0,3,2*0);
      len f = 0 by Def7;
      then f = {};
      hence Sum f = 0|^3;
    end;
A2: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat such that
A3:   P[n];
      set f = powersFS(-(n+1),3,2*(n+1));
      set g = powersFS(-n,3,2*n);
      set a = (-n) to_power 3;
      set b = (n+1) to_power 3;
A4:   Sum(<*a*>^g) = a + Sum g by RVSUM_1:76;
A5:   n|^3 = n*n*n & (n+1)|^3 = (n+1)*(n+1)*(n+1) &
      (-n) to_power 3 = (-n)*(-n)*(-n) & (n+1) to_power 3 = (n+1)*(n+1)*(n+1)
      by POLYEQ_5:2;
      f = <*a*> ^ g ^ <*b*> by Th47;
      hence Sum f = Sum(<*a*>^g) + b by RVSUM_1:74
      .= (n+1)|^3 by A3,A4,A5;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
