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;

theorem Th76:
 for I being finite initial NAT-defined Function, J being Function
 holds dom Shift(I,n) misses dom Shift(J,n+card I)
proof let I be finite initial NAT-defined Function, J be Function;
  assume
A1: dom Shift(I,n) meets dom Shift(J,n+card I);
  dom Shift(J,n+card I) = { l+(n+card I): l in dom J } by VALUED_1:def 12;
  then consider x being object such that
A2: x in dom Shift(I,n) and
A3: x in { l+(n+card I): l in dom J } by A1,XBOOLE_0:3;
 dom Shift(I,n) = { m+n:m in dom I } by VALUED_1:def 12;
  then consider m such that
A4: x = m+n and
A5: m in dom I by A2;
  consider l such that
A6: x = l+(n+card I) and
  l in dom J by A3;
  m < card I by A5,Lm1;
  hence contradiction by NAT_1:11,A4,A6,XREAL_1:6;
end;
