reserve X,Y,Z,x,y,y1,y2 for set,
  D for non empty set,
  k,n,n1,n2,m2,m1 for Nat,

  L,K,M,N for Cardinal,
  f,g for Function;
reserve r for Real;
reserve p,q for FinSequence,
  k,m,n,n1,n2,n3 for Nat;
reserve f,f1,f2 for Function,
  X1,X2 for set;

theorem
  not X is finite & Y is finite & Y <> {} implies [:X,Y:],X
  are_equipotent & card [:X,Y:] = card X
proof
  assume that
A1: not X is finite and
A2: Y is finite & Y <> {};
  card Y c= card X & 0 in card Y by A1,A2,ORDINAL3:8;
  then (card X)*`(card Y) = card X by A1,Th16;
  then card [:card X,card Y:] = card X by CARD_2:def 2;
  then card [:X,Y:] = card X by CARD_2:7;
  hence thesis by CARD_1:5;
end;
