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
  K in L & M in N implies K*`M in L*`N & M*`K in L*`N
proof
A1: for K,L,M,N st K in L & M in N & L c= N holds K*`M in L*`N
  proof
    let K,L,M,N;
    assume that
A2: K in L and
A3: M in N and
A4: L c= N;
A5: now
      assume
A6:   N is finite;
      then reconsider N as finite Cardinal;
      reconsider L,M,K as finite Cardinal by A2,A3,A4,A6,CARD_3:92;
A7:   card Segm N = N;
   card Segm M = M;
      then card M < card N by A3,A7,NAT_1:41;
      then
A8:   card K * card M <= card K * card N by XREAL_1:64;
A9:  card Segm L = L;
A10:  L*`N = card Segm(card L * card N) by CARD_2:39;
  card Segm K = K;
      then card K < card L by A2,A9,NAT_1:41;
      then card K * card N < card L * card N by A3,XREAL_1:68;
      then
A11:  card K * card M < card L * card N by A8,XXREAL_0:2;
      K*`M = card Segm(card K * card M) by CARD_2:39;
      hence thesis by A10,A11,NAT_1:41;
    end;
A12: 0 in L by A2,ORDINAL3:8;
    now
      assume
A13:  not N is finite;
      then
A14:  L*`N = N by A4,A12,Th16;
A15:  omega c= N by A13,CARD_3:85;
A16:  now
        assume K is finite & M is finite;
        then reconsider K,M as finite Cardinal;
        K*`M = card (card K * card M) by CARD_2:39
          .= (card K * card M);
        hence thesis by A14,A15;
      end;
      K c= M & (M is finite or not M is finite) or M c= K & (K is finite
      or not K is finite);
      then
      K is finite & M is finite or K*`M c= M or K*`M c= K & K in N by A2,A4
,Th17;
      hence thesis by A3,A14,A16,ORDINAL1:12;
    end;
    hence thesis by A5;
  end;
  L c= N or N c= L;
  hence thesis by A1;
end;
