reserve X,Y,Z,x,y,y1,y2 for set,
  D for non empty set,
  k,n,n1,n2,m2,m1 for Nat,

  L,K,M,N for Cardinal,
  f,g for Function;
reserve r for Real;
reserve p,q for FinSequence,
  k,m,n,n1,n2,n3 for Nat;
reserve f,f1,f2 for Function,
  X1,X2 for set;

theorem Th16:
  not M is finite & 0 in N & (N c= M or N in M) implies M*`N = M & N*`M = M
proof
A1: 1*`M = M by CARD_2:21;
  assume not M is finite;
  then
A2: M*`M = M by Th15;
  assume 0 in N;
  then 1 c= N by CARD_2:68;
  then
A3: 1*`M c= N*`M by CARD_2:90;
  assume N c= M or N in M;
  then N*`M c= M*`M by CARD_2:90;
  hence thesis by A2,A3,A1;
end;
