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 D be non empty set, F,G be Function st rng F c= D & rng G c= D &
  for d be Element of D holds card Coim(F,d) = card Coim(G,d)
  holds F,G are_fiberwise_equipotent
proof
  let D be non empty set, F,G be Function;
  assume that
A1: rng F c= D and
A2: rng G c= D;
  assume
A3: for d be Element of D holds card Coim(F,d) = card Coim(G,d);
  let x be object;
  per cases;
  suppose not x in rng F; then
A4: Coim(F,x) = {} by Lm3;
  now assume
A5:   x in rng G;
      then reconsider d = x as Element of D by A2;
   card Coim(G,d) = card Coim(F,d) by A3; then
A6:  G"{x} = {} by A4,CARD_1:5,26;
      consider y being object such that
A7:  y in dom G and
A8:  G.y = x by A5,FUNCT_1:def 3;
  G.y in {x} by A8,TARSKI:def 1;
      hence contradiction by A6,A7,FUNCT_1:def 7;
    end;
    hence thesis by A4,Lm3;
  end;
  suppose x in rng F;
    then reconsider d = x as Element of D by A1;
    card Coim(F,d) = card Coim(G,d) by A3;
    hence thesis;
  end;
end;
