theorem Th89:
  K in L & M in N or K c= L & M in N or K in L & M c= N or K c= L & M c= N
  implies K*`M c= L*`N & M*`K c= L*`N
proof
  assume that
A1: K in L & M in N or K c= L & M in N or K in L & M c= N or K c= L & M c=
  N;
A2: K c= L by A1,CARD_1:3;
A3: M c= N by A1,CARD_1:3;
A4: K*`M = card [:K,M:];
A5: [:K,M:] c= [:L,N:] by A2,A3,ZFMISC_1:96;
  hence K*`M c= L*`N by CARD_1:11;
  thus thesis by A4,A5,CARD_1:11;
end;
