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

theorem Th3:
  for f,g be FinSequence st
  f,g are_fiberwise_equipotent holds len f = len g & dom f = dom g
proof
  let f,g be FinSequence;
A1: dom f = Seg len f by FINSEQ_1:def 3;
  assume f,g are_fiberwise_equipotent;
  then
A2: card(f"(rng f)) = card(g"(rng f)) & rng f = rng g by CLASSES1:75,78;
A3: Seg len g = dom g by FINSEQ_1:def 3;
  thus len f = card(Seg len f) by FINSEQ_1:57
    .= card(g"(rng g)) by A1,A2,RELAT_1:134
    .= card(Seg len g) by A3,RELAT_1:134
    .= len g by FINSEQ_1:57;
  hence thesis by A1,FINSEQ_1:def 3;
end;
