reserve a,b,i,j,k,l,m,n for Nat;

theorem SFX:
  for f be complex-valued FinSequence, x be Complex holds
    Sum (f+x) = Sum f + x*(len f)
   proof
     let f be complex-valued FinSequence, x be Complex;
     reconsider x as Element of COMPLEX by XCMPLX_0:def 2;
     set k=len f, g = ((len f)|->x);
     (f null {}) is k-element; then
     reconsider f as k-element complex-valued FinSequence;
     reconsider h = f+x as FinSequence of COMPLEX by RVSUM_1:146;
     dom f = dom (f+x) by VALUED_1:def 2; then
     len f = len (f+x) by FINSEQ_3:29; then
     h null {} is k-element; then
     reconsider h as k-element FinSequence of COMPLEX;
     Sum (f+x) = Sum (f + g) by SL
     .= Sum f + Sum ((len f) |-> x) by RVSUM_2:40
     .= Sum f + (len f)*x by RVSUM_2:36;
     hence thesis;
   end;
