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;

theorem Th47:
  for p1 being FinSequence, p2 being FinSubsequence st len p1 <= i holds
  dom p1 misses dom Shift(p2,i)
proof
  let p1 be FinSequence, p2 be FinSubsequence;
  assume
A1: len p1 <= i;
A2: dom p1 = Seg len p1 by FINSEQ_1:def 3
    .= {k where k is Nat: 1 <= k & k <= len p1};
A3: dom Shift(p2,i) = {k+i where k is Nat: k in dom p2} by Def12;
  not ex x being object st x in dom p1 /\ dom Shift(p2,i)
  proof
    given x being object such that
A4: x in dom p1 /\ dom Shift(p2,i);
A5: x in dom p1 by A4,XBOOLE_0:def 4;
A6: x in dom Shift(p2,i) by A4,XBOOLE_0:def 4;
A7: ex k1 being Nat st ( x = k1)&( 1 <= k1)&( k1 <= len p1) by A2,A5;
    consider k2 being Nat such that
A8: x = k2+i and
A9: k2 in dom p2 by A3,A6;
    consider n being Nat such that
A10: dom p2 c= Seg n by FINSEQ_1:def 12;
A11: k2 in Seg n by A9,A10;
A12:  ex m being Nat st k2 = m & 1 <= m & m <= n by A11;
    reconsider x as Element of NAT by A4;
    len p1 + k2 <= i+k2 by A1,XREAL_1:7;
    then (len p1 + k2) - k2 < x - 0 by A8,A12,XREAL_1:15;
    hence contradiction by A7;
  end;
  hence dom p1 /\ dom Shift(p2,i) = {} by XBOOLE_0:def 1;
end;
