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 Th40:
  for q being FinSubsequence ex ss being FinSubsequence st
  dom ss = dom q & rng ss = dom Shift(q,i) &
  (for k st k in dom q holds ss.k = i+k) & ss is one-to-one
proof
  let q be FinSubsequence;
  consider n being Nat such that
A1: dom q c= Seg n by FINSEQ_1:def 12;
  defpred P[object,object] means ex k st k = $1 & $2 = i+k;
A2: for x being object st x in dom q ex y being object st P[x,y]
  proof
    let x be object;
    assume x in dom q;
    then reconsider x as Element of NAT;
    take i+x;
    thus thesis;
  end;
  consider f being Function such that
A3: dom f = dom q and
A4: for x being object st x in dom q holds P[x, f.x] from CLASSES1:sch 1(A2);
  reconsider ss = f as FinSubsequence by A1,A3,FINSEQ_1:def 12;
A5: dom Shift(q,i) = {k+i where k is Nat: k in dom q} by Def12;
A6: rng ss = dom Shift(q,i)
  proof
    thus rng ss c= dom Shift(q,i)
    proof
      let y be object;
      assume y in rng ss;
      then consider x being object such that
A7:   x in dom ss and
A8:   y = ss.x by FUNCT_1:def 3;
      ex k1 st ( k1 = x)&( ss.x = i+k1) by A3,A4,A7;
      hence thesis by A3,A5,A7,A8;
    end;
    let y be object;
    assume y in dom Shift(q,i);
    then consider k2 being Nat such that
A9: y = k2+i and
A10: k2 in dom q by A5;
    ex k3 being Element of NAT st ( k3 = k2)&( ss.k2 = i+k3)
    by A4,A10;
    hence thesis by A3,A9,A10,FUNCT_1:def 3;
  end;
A11: for k st k in dom q holds ss.k = i+k
  proof
    let k;
    assume k in dom q;
    then ex k1 st ( k1 = k)&( ss.k = i+k1) by A4;
    hence thesis;
  end;
  for x1,x2 being object
   st x1 in dom ss & x2 in dom ss & ss.x1 = ss.x2 holds x1 = x2
  proof
    let x1,x2 be object;
    assume that
A12: x1 in dom ss and
A13: x2 in dom ss and
A14: ss.x1 = ss.x2;
A15: ex k1 st ( k1 = x1)&( ss.x1 = i+k1) by A3,A4,A12;
    ex k2 st ( k2 = x2)&( ss.x2 = i+k2) by A3,A4,A13;
    hence thesis by A14,A15;
  end;
  then ss is one-to-one;
  hence thesis by A3,A6,A11;
end;
