reserve X for set;
reserve k,m,n for Nat;
reserve i for Integer;
reserve a,b,c,d,e,g,p,r,x,y for Real;
reserve z for Complex;

theorem
  for f being complex-valued FinSequence, c being Complex holds
  Sum(c+f) = (c*len f) + Sum f
  proof
    let f be complex-valued FinSequence;
    let c be Complex;
    defpred P[complex-valued FinSequence] means
    Sum(c+$1) = (c*len $1) + Sum $1;
A1: P[<*>COMPLEX];
A2: for p being FinSequence of COMPLEX, x being Element of COMPLEX
    st P[p] holds P[p^<*x*>]
    proof
      let p be FinSequence of COMPLEX, x be Element of COMPLEX such that
A3:   Sum(c+p) = (c*len p) + Sum p;
      set g = p^<*x*>;
A4:   len <*x*> = 1 by FINSEQ_1:39;
A5:   len g = len p + len <*x*> by FINSEQ_1:22;
A6:   c+<*x*> = <*c+x*> by Th2;
A7:   Sum g = Sum p + Sum <*x*> by RVSUM_2:32;
      c+g = (c+p)^(c+<*x*>) by Th3;
      hence Sum(c+g) = Sum(c+p) + Sum(c+<*x*>) by RVSUM_2:32
      .= (c*len g) + Sum g by A3,A4,A5,A6,A7;
    end;
A8: for p being FinSequence of COMPLEX holds P[p] from FINSEQ_2:sch 2(A1,A2);
    f is FinSequence of COMPLEX by RVSUM_1:146;
    hence thesis by A8;
  end;
