reserve x,y for set,
  n,m for Nat,
  r,s for Real;
reserve f, g for Function;

theorem
  for f, g st rng f = rng g & f is one-to-one & g is one-to-one holds
  f,g are_fiberwise_equipotent
proof
  let f, g be Function such that
A1: rng f = rng g and
A2: f is one-to-one and
A3: g is one-to-one;
  let x be object;
  per cases;
  suppose
A4: x in rng f;
    then card Coim(f,x) = 1 by A2,FINSEQ_4:73;
    hence thesis by A1,A3,A4,FINSEQ_4:73;
  end;
  suppose
A5: not x in rng f;
    then card(f"{x}) = 0 by CARD_1:27,FUNCT_1:72;
    hence thesis by A1,A5,CARD_1:27,FUNCT_1:72;
  end;
end;
