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 Th3:
  for f be sequence of Y holds (for k1 holds x in f.(n+k1)) iff
  for Z st Z in {f.k2 : n <= k2} holds x in Z
proof
  let f be sequence of Y;
  set Z = {f.k2 : n <= k2};
  now
    assume
A1: for k1 holds x in f.(n+k1);
    now
      let Z1 be set;
      assume Z1 in Z;
      then consider k1 being Nat such that
A2:   Z1=f.k1 and
A3:   n <= k1;
      ex m be Nat st k1 = n + m by A3,NAT_1:10;
      then consider k2 being Nat such that
A4:   Z1=f.(n + k2) by A2;
      thus x in Z1 by A1,A4;
    end;
    hence for Z1 being set st Z1 in {f.k2 : n <= k2} holds x in Z1;
  end;
  hence thesis by Th1;
end;
