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 Th34:
  X misses Y implies card (X \/ Y) = card X +` card Y
proof
  assume
A1: X misses Y;
  X,[:X,{0}:] are_equipotent & [:X,{0}:],[:card X,{0}:] are_equipotent
     by CARD_1:69
,Th6;
  then
A2: X,[:card X,{0}:] are_equipotent by WELLORD2:15;
  Y,[:Y,{1}:] are_equipotent & [:Y,{1}:],[:card Y,{1}:] are_equipotent
   by CARD_1:69,Th6;
  then
A3: Y,[:card Y,{1}:] are_equipotent by WELLORD2:15;
  [:card X,{0}:] misses [:card Y,{1}:] by Lm4;
  then X \/ Y,[:card X,{0}:] \/ [:card Y,{1}:] are_equipotent by A1,A2,A3,
CARD_1:31;
  hence card (X \/ Y) = card ([:card X,{0}:] \/ [:card Y,{1}:]) by CARD_1:5
    .= card X +` card Y by Th9;
end;
