reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;
reserve i for Nat;
reserve m for Nat,
        D for non empty set;
reserve l for Nat;
reserve M for Nat;
reserve m,n for Nat;
reserve x1,x2,x3,x4 for object;
reserve e,u for object;

theorem Th84:
 dom Shift(<%e%>,card p) = {card p}
proof
  for u holds u in dom Shift(<%e%>,card p) iff u = card p
   proof let u;
    thus u in dom Shift(<%e%>,card p) implies u = card p
     proof
      assume u in dom Shift(<%e%>,card p);
       then u in { m+card p where m is Nat:m in dom <%e%> } by VALUED_1:def 12;
       then consider m being Nat such that
A1:    u = m+card p and
A2:    m in dom <%e%>;
       m = 0 by A2,TARSKI:def 1;
      hence u = card p by A1;
     end;
     0 in 1 by CARD_1:49,TARSKI:def 1;
     then 0 in dom <%e%> by Def4;
     then 0+card p in dom Shift(<%e%>,card p) by VALUED_1:24;
     hence thesis;
   end;
  hence thesis by TARSKI:def 1;
end;
