reserve m,j,p,q,n,l for Element of NAT;
reserve e1,e2 for ExtReal;
reserve i for Nat,
        k,k1,k2,j1 for Element of NAT,
        x,x1,x2,y for set;

theorem Th44:
  for p being FinSequence st k in dom p holds (Seq Shift(p,i)).k = p.k
proof
  let p be FinSequence;
  assume
A1: k in dom p;
  then
A2: k in dom Seq Shift(p,i) by Th42;
  (Shift(p,i)*(Sgm dom Shift(p,i))).k
  = Shift(p,i).((Sgm dom Shift(p,i)).k) by A2,FUNCT_1:12
    .= Shift(p,i).(i+k) by A1,Th43;
  hence thesis by A1,Def12;
end;
