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;

theorem
 for I being finite initial NAT-defined Function, J being Function
 holds dom I misses dom Shift(J,card I)
proof let I be finite initial NAT-defined Function, J be Function;
  assume
A1: dom I meets dom Shift(J,card I);
  dom Shift(J,card I) = { l+card I: l in dom J } by VALUED_1:def 12;
  then consider x being object such that
A2: x in dom I and
A3: x in { l+card I: l in dom J } by A1,XBOOLE_0:3;
  consider l such that
A4: x = l+card I and
  l in dom J by A3;
  thus contradiction by NAT_1:11,A2,A4,Lm1;
end;
