
theorem Th12:
  for M,N being Cardinal, f being Function holds
    (M c= card dom f & for x being object st x in dom f
      holds N c= card (f.x)) implies M*`N c= Sum Card f
proof
  let M,N being Cardinal, f being Function;
  assume that
A1: M c= card dom f and
A2: for x being object st x in dom f holds N c= card (f.x);
A3: dom Card f = dom f by CARD_3:def 2;
A4: dom(dom f --> N) = dom f;
  now
    let x be object;
    assume
A5: x in dom Card f;
    then
A6: (Card f).x = card (f.x) by A3,CARD_3:def 2;
    (dom f --> N).x = N by A3,A5,FUNCOP_1:7;
    hence (dom f --> N).x c= (Card f).x by A2,A3,A5,A6;
  end;
  then
A7: Sum(dom f --> N) c= Sum Card f by A3,A4,CARD_3:30;
A8: [:M,N:] c= [:card dom f,N:] by A1,ZFMISC_1:95;
  Sum(dom f --> N) = card Union disjoin (dom f --> N) by CARD_3:def 7
    .= card [:N,dom f:] by CARD_3:25
    .= card [:N,card dom f:] by CARD_2:7
    .= card [:card dom f,N:] by CARD_2:4;
  then card [:M,N:] c= Sum(dom f --> N) by A8,CARD_1:11;
  then card [:M,N:] c= Sum Card f by A7,XBOOLE_1:1;
  hence thesis by CARD_2:def 2;
end;
