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 Th39:
  for p being FinSequence
  holds  dom Shift(p,i) = {j1 where j1 is Nat: i+1 <= j1 & j1 <= i+(len p)}
proof
  let p be FinSequence;
A1: dom p = Seg len p by FINSEQ_1:def 3
    .= {k where k is Nat: 1 <= k & k <= len p};
  set X = {j1 where j1 is Nat: i+1 <= j1 & j1 <= i+(len p)};
A2: dom Shift(p,i) = {k1+i where k1 is Nat: k1 in dom p} by Def12;
  thus dom Shift(p,i) c= X
  proof
    let x be object;
    assume x in dom Shift(p,i);
    then consider k1 being Nat such that
A3: x = k1+i and
A4: k1 in dom p by A2;
    consider k being Nat such that
A5: k1 = k and
A6: 1 <= k and
A7: k <= len p by A1,A4;
A8: i+1 <= i+k by A6,XREAL_1:7;
    i+k <= i+(len p) by A7,XREAL_1:7;
    hence thesis by A3,A5,A8;
  end;
  let x be object;
  assume x in X;
  then consider j1 being Nat such that
A9: x = j1 and
A10: i+1 <= j1 and
A11: j1 <= i+(len p);
  i+(0 qua Nat) <= i+1 by XREAL_1:7;
  then consider k2 being Nat such that
A12: j1 = i+k2 by A10,NAT_1:10,XXREAL_0:2;
A13: 1 <= k2 by A10,A12,XREAL_1:6;
  k2 <= len p by A11,A12,XREAL_1:6;
  then k2 in dom p by A1,A13;
  hence thesis by A2,A9,A12;
end;
