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
 for X,Y being finite set st X c= Y & card X=card Y
  holds X=Y
proof let X,Y be finite set such that
A1: X c= Y and
A2: card X=card Y;
  card (Y\X)=card Y-card X by A1,Th43;
  then Y\X={} by A2;
  then Y c= X by XBOOLE_1:37;
  hence thesis by A1;
end;
