reserve x, x1, x2, y, z, X9 for set,
  X, Y for finite set,
  n, k, m for Nat,
  f for Function;

theorem Th14:
  for z,x,y being object holds
  x<>y & not z in X implies card Choose(X\/{z},k+1,x,y)= card
  Choose(X,k+1,x,y)+ card Choose(X,k,x,y)
proof let z,x,y be object;
  assume that
A1: x<>y and
A2: not z in X;
  set F2={f where f is Function of X\/{z},{x,y}:card (f"{x})=k+1 & f.z=y};
  set F1={f where f is Function of X\/{z},{x,y}:card (f"{x})=k+1 & f.z=x};
A3: F1 \/F2= Choose(X\/{z},k+1,x,y) by A1,Lm1;
  F1 c= F1\/F2 & F2 c= F1\/F2 by XBOOLE_1:7;
  then reconsider F1,F2 as finite set by A3;
A4: card F1=card Choose(X,k,x,y) by A2,Th12;
  card (F1 \/ F2) = card F1 + card F2 & card F2=card Choose(X,k+1,x,y)
  by A1,A2
,Lm1,Th13,CARD_2:40;
  hence thesis by A1,A4,Lm1;
end;
