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
  (card X c= M & for Y st Y in X holds card Y c= N) implies card union X c=
  M*`N
proof
  assume that
A1: card X c= M and
A2: for Y st Y in X holds card Y c= N;
  now
    let x;
    assume x in dom id X;
    then (id X).x in X by FUNCT_1:18;
    hence card ((id X).x) c= N by A2;
  end;
  then card Union id X c= M*`N by A1,Th85;
  hence thesis;
end;
