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 Th18:
  (K+`M)+`N = K+`(M+`N)
proof
A1: card(K+`M+`N) = K+`M+`N;
A2: K+`M+`N,[:K+`M,{0}:] \/ [:N,{2}:] are_equipotent by Th9;
  [:M,{1}:] misses [:N,{2}:] by Lm4;
  then
A3: [:M,{1}:] /\ [:N,{2}:] = {};
  [:K,{0}:] misses [:N,{2}:] by Lm4;
  then [:K,{0}:] /\ [:N,{2}:] = {};
  then ([:K,{0}:] \/ [:M,{1}:]) /\ [:N,{2}:] = {} \/ {} by A3,XBOOLE_1:23
    .= {};
  then
A4: ([:K,{0}:] \/ [:M,{1}:]) misses [:N,{2}:];
  K+`M,[:K,{0}:] \/ [:M,{1}:] are_equipotent & K+`M,[:K+`M,{0}:]
  are_equipotent by CARD_1:69,Th9;
  then
A5: [:K+`M,{0}:],[:K,{0}:] \/ [:M,{1}:] are_equipotent by WELLORD2:15;
  [:K,{0}:] misses [:N,{2}:] by Lm4;
  then
A6: [:K,{0}:] /\ [:N,{2}:] = {};
  [:K,{0}:] misses [:M,{1}:] by Lm4;
  then [:K,{0}:] /\ [:M,{1}:] = {};
  then [:K,{0}:] /\ ([:M,{1}:] \/ [:N,{2}:]) = {} \/ {} by A6,XBOOLE_1:23
    .= {};
  then
A7: [:K,{0}:] misses ([:M,{1}:] \/ [:N,{2}:]);
  M+`N,[:M,{1}:] \/ [:N,{2}:] are_equipotent & M+`N,[:M+`N,{2}:]
  are_equipotent by CARD_1:69,Th9;
  then
A8: [:M+`N,{2}:],[:M,{1}:] \/ [:N,{2}:] are_equipotent by WELLORD2:15;
  [:K,{0}:] misses [:M+`N,{2}:] by Lm4;
  then
A9: [:K,{0}:] \/ ([:M,{1}:] \/ [:N,{2}:]),[:K,{0}:] \/ [:M+`N,{2}:]
  are_equipotent by A7,A8,CARD_1:31;
  [:K+`M,{0}:] misses [:N,{2}:] by Lm4;
  then
A10: [:K+`M,{0}:] \/ [:N,{2}:], [:K,{0}:] \/ [:M,{1}:] \/ [:N,{2}:]
  are_equipotent by A4,A5,CARD_1:31;
  [:K,{0}:] \/ ([:M,{1}:] \/ [:N,{2}:]) = [:K,{0}:] \/ [:M,{1}:] \/ [:N,{
  2}:] by XBOOLE_1:4;
  then [:K+`M,{0}:] \/ [:N,{2}:],[:K,{0}:] \/ [:M+`N,{2}:] are_equipotent by
A10,A9,WELLORD2:15;
  then
A11: K+`M+`N,[:K,{0}:] \/ [:M+`N,{2}:] are_equipotent by A2,WELLORD2:15;
  [:K,{0}:] \/ [:M+`N,{2}:],K+`(M+`N) are_equipotent by Th9;
  then K+`M+`N,K+`(M+`N) are_equipotent by A11,WELLORD2:15;
  hence thesis by A1,CARD_1:def 2;
end;
