reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;
reserve x1,x2,x3,x4,x5,x6,x7,x8 for object;
reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve n,k for Nat;

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;
