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 Th85:
  (card dom f c= M & for x st x in dom f holds card (f.x) c= N) implies
  card Union f c= M*`N
proof
  assume that
A1: card dom f c= M and
A2: for x st x in dom f holds card (f.x) c= N;
A3: card Union f c= Sum Card f by CARD_3:39;
A4: dom Card f = dom f by CARD_3:def 2;
A5: dom(dom f --> N) = dom f by FUNCOP_1:13;
  now
    let x be object;
    assume
A6: x in dom Card f;
    then
A7: (Card f).x = card (f.x) by A4,CARD_3:def 2;
    (dom f --> N).x = N by A4,A6,FUNCOP_1:7;
    hence (Card f).x c= (dom f --> N).x by A2,A4,A6,A7;
  end;
  then Sum Card f c= Sum(dom f --> N) by A4,A5,CARD_3:30;
  then
A8: card Union f c= Sum(dom f --> N) by A3;
A9: [:card dom f,N:] c= [:M,N:] by A1,ZFMISC_1:95;
  Sum(dom f --> N) = card [:N,dom f:] by CARD_3:25
    .= card [:N,card dom f:] by Th6
    .= card [:card dom f,N:] by Th4;
  then
A10: Sum(dom f --> N) c= card [:M,N:] by A9,CARD_1:11;
  thus thesis by A8,A10;
end;
