reserve n,m,k,k1,k2,i,j for Nat;
reserve x,y,z for object,X,Y,Z for set;
reserve A for Subset of X;
reserve B,A1,A2,A3 for SetSequence of X;
reserve Si for SigmaField of X;
reserve S,S1,S2,S3 for SetSequence of Si;

theorem Th6:
  for Y being non empty set for f being sequence of Y holds rng
  (f ^\ k) = {f.n: k <= n}
proof
  let Y be non empty set;
  let f be sequence of Y;
  reconsider f1 = f ^\ k as sequence of Y;
  rng f1 = {f.m: k <= m}
  proof
    set Z = {f.m : k <= m};
A1: dom f1 = NAT by FUNCT_2:def 1;
A2: rng f1 c= Z
    proof
      let y be object;
      assume y in rng f1;
      then consider m1 be object such that
A3:   m1 in NAT and
A4:   y = f1.m1 by A1,FUNCT_1:def 3;
      reconsider m1 as Element of NAT by A3;
      y = f.(k+m1) by A4,NAT_1:def 3;
      hence thesis by Th1;
    end;
    Z c= rng f1
    proof
      let x be object;
      assume x in Z;
      then consider k1 being Nat such that
A5:   x = f.k1 and
A6:   k <= k1;
      consider k2 being Nat such that
A7:   k1 = k + k2 by A6,NAT_1:10;
      k2 in NAT & x = f1.k2 by A5,A7,NAT_1:def 3,ORDINAL1:def 12;
      hence thesis by FUNCT_2:4;
    end;
    hence thesis by A2,XBOOLE_0:def 10;
  end;
  hence thesis;
end;
