reserve a,b,m,x,n,l,xi,xj for Nat,
  t,z for Integer;

theorem
  for F,G being integer-valued FinSequence holds
  (a*(F ^ G)) mod m = ((a*F) mod m) ^ ((a*G) mod m)
proof
  let F,G be integer-valued FinSequence;
   set FG = F^G;
A4: dom (a*(F ^ G)) = dom (F ^ G) by VALUED_1:def 5;
A5: dom (a*G) = dom G by VALUED_1:def 5;
    then
A6: dom ((a*G) mod m) = dom G by Def1;
A7: dom (a*F) = dom F by VALUED_1:def 5;
    then
A8: dom ((a*F) mod m) = dom F by Def1;
A9: for x being object st x in dom (F ^ G)
    holds ((a*(F ^ G)) mod m).x = (((a*F) mod m) ^ ((a*G) mod m)).x
  proof
    set H = F^G;
    let x be object;
    assume
A10: x in dom (F ^ G);
    now
      per cases by A10,FINSEQ_1:25;
      suppose
A11:     x in dom F;
A12:     ((a*(F ^ G)) mod m).x = ((a*(F ^ G)).x) mod m by A4,A10,Def1
          .= (a*H.x) mod m by VALUED_1:6
          .= (a*(F.x)) mod m by A11,FINSEQ_1:def 7;
        (((a*F) mod m) ^ ((a*G) mod m)).x = ((a*F) mod m).x by A8,A11,
FINSEQ_1:def 7
          .= ((a*F).x) mod m by A7,A11,Def1
          .= (a*(F.x)) mod m by VALUED_1:6;
        hence ((a*(F ^ G)) mod m).x = (((a*F) mod m) ^ ((a*G) mod m)).x by A12;
      end;
      suppose
        ex n being Nat st n in dom G & x=len F+n;
        then consider n being Element of NAT such that
A13:    n in dom G and
A14:    x=len F+n;
A15:    ((a*(F ^ G)) mod m).x = ((a*(F ^ G)).x) mod m by A4,A10,Def1
          .= (a*H.x) mod m by VALUED_1:6
          .= (a*G.n) mod m by A13,A14,FINSEQ_1:def 7;
        len ((a*F) mod m) = len(a*F) by Def2
          .= len F by NEWTON:2;
        then
        (((a*F) mod m) ^ ((a*G) mod m)).x = ((a*G) mod m).n by A6,A13,A14,
FINSEQ_1:def 7
          .= ((a*G).n) mod m by A5,A13,Def1
          .= (a*G.n) mod m by RVSUM_1:44;
        hence ((a*(F ^ G)) mod m).x = (((a*F) mod m) ^ ((a*G) mod m)).x by A15;
      end;
    end;
    hence thesis;
  end;
A16: dom ((a*(F ^ G)) mod m) = dom (a*(F ^ G)) by Def1
    .= dom (F ^ G) by VALUED_1:def 5;
  dom (((a*F) mod m) ^ ((a*G) mod m)) = Seg(len(((a*F) mod m) ^ ((a*G) mod
  m))) by FINSEQ_1:def 3
    .= Seg(len((a*F) mod m)+len((a*G) mod m)) by FINSEQ_1:22
    .= Seg(len(a*F)+len((a*G) mod m)) by Def2
    .= Seg(len(a*F)+len(a*G)) by Def2
    .= Seg(len(F)+len(a*G)) by NEWTON:2
    .= Seg(len(F)+len(G)) by NEWTON:2
    .= Seg(len(F ^ G)) by FINSEQ_1:22
    .= dom(F ^ G) by FINSEQ_1:def 3;
  hence thesis by A16,A9;
end;
