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 Th42:
  for p being FinSequence holds dom p = dom Seq Shift(p,i)
proof
  let p be FinSequence;
A1: rng Sgm dom Shift(p,i) = dom Shift(p,i) by FINSEQ_1:50;
A2: dom p = Seg len p by FINSEQ_1:def 3;
A3: dom Sgm dom Shift(p,i) = Seg len Sgm dom Shift(p,i) by FINSEQ_1:def 3;
A4: len Sgm dom Shift(p,i) = card dom Shift(p,i) by FINSEQ_3:39;
  card dom Shift(p,i) = card Shift(p,i) by CARD_1:62;
  then card dom Shift(p,i) = len p by Th41;
  hence thesis by A1,A2,A3,A4,RELAT_1:27;
end;
