reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th44:
  for f,g be FinSequence st f^g is complex-functions-valued holds
       f is complex-functions-valued
     & g is complex-functions-valued
proof
  let f,g be FinSequence such that A1:f^g is complex-functions-valued;
  A2:now let x be object;
    A3:dom f c= dom (f^g) by FINSEQ_1:26;
    assume x in dom f;
    then f.x = (f^g).x & x in dom (f^g) by A3,FINSEQ_1:def 7;
    hence f.x is complex-valued Function by A1;
  end;
  now let x be object;
    assume A4:x in dom g;
    then reconsider xx=x as Nat;
    g.x = (f^g).(xx+len f) by A4,FINSEQ_1:def 7;
    hence g.x is complex-valued Function by A1;
  end;
  hence thesis by A2,VALUED_2:def 26;
end;
