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;
reserve p1,p2 for FinSequence;
reserve q,q1,q2,q3,q4 for FinSubsequence,
        p1,p2 for FinSequence;
reserve l1 for Nat,
        j2 for Element of NAT;

theorem Th57:
  for q being FinSubsequence st
  k in dom Seq q holds (Seq Shift(q,i)).k = (Seq q).k
proof
  let q be FinSubsequence;
  assume
A1: k in dom Seq q;
  then consider j such that
A2: j = (Sgm dom q).k and
A3: (Sgm dom Shift(q,i)).k = i+j by Th56;
A4: rng Sgm dom q = dom q by FINSEQ_1:50;
A5: dom Seq q = dom Seq Shift(q,i) by Th55
    .= dom (Shift(q,i)*(Sgm dom Shift(q,i)));
A6: dom Seq q = dom (q*Sgm dom q)
    .= dom Sgm dom q by A4,RELAT_1:27;
  then j in rng Sgm dom q by A1,A2,FUNCT_1:def 3;
  then
A7: j in dom q by FINSEQ_1:50;
  (Seq Shift(q,i)).k = (Shift(q,i)*(Sgm dom Shift(q,i))).k
    .= Shift(q,i).((Sgm dom Shift(q,i)).k)
  by A1,A5,FUNCT_1:12
    .= q.j by A3,A7,Def12
    .= (q*Sgm dom q).k by A1,A2,A6,FUNCT_1:13
    .= (Seq q).k;
  hence thesis;
end;
