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 Th20:
 for n being Nat holds
  x in (superior_setsequence B).n iff
   ex k being Nat st x in B.(n + k)
proof let n be Nat;
  set Y = {B.k : n <= k};
A1: (superior_setsequence B).n = union {B.k : n <= k} by Def3;
  thus x in (superior_setsequence B).n
   implies ex k being Nat st x in B.(n + k)
  proof
    assume x in (superior_setsequence B).n;
    then consider Y1 being set such that
A2: x in Y1 and
A3: Y1 in Y by A1,TARSKI:def 4;
    consider k11 being Nat such that
A4: Y1 = B.k11 and
A5: n <= k11 by A3;
    ex k be Nat st k11 = n + k by A5,NAT_1:10;
    then consider k22 being Nat such that
A6: Y1=B.(n + k22) by A4;
    thus thesis by A2,A6;
  end;
  now
    given k1 being Nat such that
A7: x in B.(n + k1);
     B.(n+k1) in Y by Th1;
    hence x in (superior_setsequence B).n by A1,A7,TARSKI:def 4;
  end;
  hence thesis;
end;
