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

theorem
  for a be Complex, f be complex-valued FinSequence holds
    ((len f)|->a)(#)f = a(#)f
   proof
     let a be Complex, f be complex-valued FinSequence;
     len ((len f)|->a) = len f; then
     dom ((len f)|->a) = dom f by FINSEQ_3:29; then
     A1: (dom ((len f)|->a)) /\ (dom f) = dom (a(#)f) by VALUED_1:def 5; then
     A2: dom (((len f)|->a)(#)f) = dom (a(#)f) by VALUED_1:def 4;
     for x be object st x in dom (((len f)|->a)(#)f) holds
     (((len f)|->a)(#)f).x = (a(#)f).x
     proof let x be object such that
       B1: x in dom (((len f)|-> a)(#)f);
       reconsider x as non zero Nat by B1,FINSEQ_3:25;
       len ((len f)|->a) >= x >= 1 by A1,A2,B1,FINSEQ_3:25; then
       ex k be Nat st len f = x+k by NAT_1:10; then
       reconsider k = len f - x as Nat;
       (((len f)|->a)(#)f).x = ((x+k)|-> a).x*(f.x) by B1,VALUED_1:def 4
       .= (a(#)f).x by A2,B1,VALUED_1:def 5;
       hence thesis;
     end;
     hence thesis by A2,FUNCT_1:2;
   end;
