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 Th56:
  (c(#)f)|n = c(#)(f|n)
  proof
A1: len(c(#)f) = len f by COMPLSP2:3;
A2: len(c(#)(f|n)) = len(f|n) by COMPLSP2:3;
    per cases;
    suppose
A3:   n <= len f;
      then
A4:   len((c(#)f)|n) = n by A1,FINSEQ_1:59;
      hence len((c(#)f)|n) = len(c(#)(f|n)) by A2,A3,FINSEQ_1:59;
      let k such that
      1 <= k and
A5:   k <= len((c(#)f)|n);
A6:   (f|n).k = f.k by A4,A5,FINSEQ_3:112;
      thus ((c(#)f)|n).k = (c(#)f).k by A4,A5,FINSEQ_3:112
      .= c*f.k by VALUED_1:6
      .= (c(#)(f|n)).k by A6,VALUED_1:6;
    end;
    suppose
A7:   n > len f;
      then
A8:   f|n = f by FINSEQ_1:58;
      hence len((c(#)f)|n) = len(c(#)(f|n)) by A2,A7,FINSEQ_1:58;
      thus thesis by A1,A7,A8,FINSEQ_1:58;
    end;
  end;
