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;

theorem Th23:
  K*`(M+`N) = K*`M +` K*`N
proof
A1: [:card [:K,M:],{0}:],[:[:K,M:],{0}:] are_equipotent by Th6;
  M,[:M,{0}:] are_equipotent by CARD_1:69;
  then
A2: [:K,M:],[:K,[:M,{0}:]:] are_equipotent by Th7;
  [:[:K,M:],{0}:],[:K,M:] are_equipotent by CARD_1:69;
  then [:[:K,M:],{0}:],[:K,[:M,{0}:]:] are_equipotent by A2,WELLORD2:15;
  then
A3: [:card [:K,M:],{0}:],[:K,[:M,{0}:]:] are_equipotent by A1,WELLORD2:15;
A4: [:card [:K,N:],{1}:],[:[:K,N:],{1}:] are_equipotent by Th6;
  [:M,{0}:] misses [:N,{1}:] by Lm4;
  then
A5: [:K,[:M,{0}:]:] misses [:K,[:N,{1}:]:] by ZFMISC_1:104;
  N,[:N,{1}:] are_equipotent by CARD_1:69;
  then
A6: [:K,N:],[:K,[:N,{1}:]:] are_equipotent by Th7;
A7: K*`(M+`N) = card [:K,card plus(M,N):] by Th9
    .= card [:K,plus(M,N):] by Th6
    .= card ([:K,[:M,{0}:]:] \/ [:K,[:N,{1}:]:]) by ZFMISC_1:97;
  [:[:K,N:],{1}:],[:K,N:] are_equipotent by CARD_1:69;
  then [:[:K,N:],{1}:],[:K,[:N,{1}:]:] are_equipotent by A6,WELLORD2:15;
  then
A8: [:card [:K,N:],{1}:],[:K,[:N,{1}:]:] are_equipotent by A4,WELLORD2:15;
  [:card [:K,M:],{0}:] misses [:card [:K,N:],{1}:] by Lm4;
  then
  [:card [:K,M:],{0}:] \/ [:card [:K,N:],{1}:], [:K,[:M,{0}:]:] \/ [:K,[:
  N,{1}:]:] are_equipotent by A3,A8,A5,CARD_1:31;
  hence K*`(M+`N) = card([:card [:K,M:],{0}:] \/ [:card [:K,N:],{1}:]) by A7,
CARD_1:5
    .= K*`M +` K*`N by Th9;
end;
