reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;
reserve f,g for complex-valued FinSequence;

theorem Th69:
  for s,n being Nat
  for f being (s+1)-element complex-valued FinSequence holds
  f is a_solution_of_Sierp168 implies n(#)f is a_solution_of_Sierp168
  proof
    let s,n be Nat;
    let f be (s+1)-element complex-valued FinSequence such that
A1: Sum ((f|s)" ^2) = 1 / (f.(s+1))^2;
    set g = n(#)f;
A2: g.(s+1) = n*f.(s+1) by VALUED_1:6;
A3: (n(#)(f|s))^2 = (n^2)(#)((f|s)^2) by BORSUK_7:18;
A4: (f|s)" ^2 = (f|s)^2" by Th57;
A5: (n(#)(f|s))" ^2 = (n(#)(f|s))^2 " by Th57;
A6: (n^2(#)(f|s)^2) " = (1/n^2) (#) (((f|s)^2)") by Th58;
    g|s = n(#)(f|s) by Th56;
    hence Sum ((g|s)" ^2) = 1/n^2 * Sum(((f|s)^2)") by A3,A5,A6,RVSUM_2:38
    .= (1*1) / (n^2*(f.(s+1))^2) by A1,A4,XCMPLX_1:76
    .= 1 / (g.(s+1))^2 by A2;
  end;
