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 Th17:
  not M is finite & (N c= M or N in M) implies M*`N c= M & N*`M c= M
proof
  assume not M is finite & (N c= M or N in M);
  then M*`N = M or not 0 in N by Th16;
  then M*`N c= M or N = 0 & M*`0 = 0 & 0 c= M
  by CARD_2:20,ORDINAL3:8;
  hence thesis;
end;
