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
  not X is finite & Y is finite implies
  X \/ Y,X are_equipotent & card (X \/ Y) = card X
proof
  assume that
A1: not X is finite and
A2: Y is finite;
  card Y in card X by A1,A2,CARD_3:86;
  then
A3: card X +` card Y = card X by A1,Th75;
A4: card (X \/ Y) c= card X +` card Y by Th33;
  card X c= card (X \/ Y) by CARD_1:11,XBOOLE_1:7;
  then card X = card (X \/ Y) by A3,A4;
  hence thesis by CARD_1:5;
end;
