reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;
reserve n for Element of omega;
reserve e,u for set;

theorem
  for F,G be Function
  st F,G are_fiberwise_equipotent holds card dom F = card dom G
proof
  let F,G be Function;
  assume F,G are_fiberwise_equipotent;
  then card(F"(rng F)) = card(G"(rng F)) & rng F = rng G by Th75,Th78;
  hence card(dom F) = card(G"(rng G)) by RELAT_1:134
    .= card(dom G) by RELAT_1:134;
end;
