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 Th59:
  for q being FinSubsequence st
  dom q c= Seg k holds dom Shift(q,i) c= Seg (i+k)
proof
  let q be FinSubsequence;
  assume
A1: dom q c= Seg k;
A2: dom Shift(q,i) = {j+i where j is Nat: j in dom q} by Def12;
  let x be object;
  assume x in dom Shift(q,i);
  then consider j1 being Nat such that
A3: x = j1+i and
A4: j1 in dom q by A2;
  j1 in Seg k by A1,A4;
  then
A5: ex j2 being Nat st ( j1 = j2)&( 1 <= j2)&( j2 <= k);
A6: 0 qua Nat+1 <= i+j1 by A5,XREAL_1:7;
  i+j1 <= i+k by A5,XREAL_1:7;
  hence thesis by A3,A6;
end;
