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 Th87:
  for f st dom f is finite &
  for x st x in dom f holds f.x is finite holds Union f is finite
proof
  let f;
  assume that
A1: dom f is finite and
A2: for x st x in dom f holds f.x is finite;
  reconsider df = dom f as finite set by A1;
  now
    assume dom f <> {};
    then
A3: df <> {};
    defpred P[object,object] means card (f.$2) c= card (f.$1);
A4: for x,y holds P[x,y] or P[y,x];
A5: for x,y,z being object st P[x,y] & P[y,z] holds P[x,z] by XBOOLE_1:1;
    consider x such that
A6: x in df & for y st y in df holds P[x,y] from MaxFinSetElem(A3,A4,
    A5);
    reconsider fx = f.x as finite set by A2,A6;
A7: card Union f c= (card card df) *` (card (f.x)) by A6,Th85;
    card (f.x) = card card fx;
    then card Union f c= card ((card df) * (card fx)) by A7,Th38;
    hence thesis;
  end;
  hence thesis by RELAT_1:42,ZFMISC_1:2;
end;
