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

theorem Th31:
  for f,g,h be FinSequence holds
  f,g are_fiberwise_equipotent iff h^f, h^g are_fiberwise_equipotent
proof
  let f,g,h be FinSequence;
  thus f,g are_fiberwise_equipotent implies h^f, h^g are_fiberwise_equipotent
  proof
    assume
A1: f,g are_fiberwise_equipotent;
    now
      let y be object;
      card Coim(f,y) = card Coim(g,y) by A1;
      hence card Coim(h^f,y) = card(g"{y}) + card(h"{y}) by FINSEQ_3:57
        .= card Coim(h^g,y) by FINSEQ_3:57;
    end;
    hence thesis;
  end;
  assume
A2: h^f,h^g are_fiberwise_equipotent;
  now
    let x be object;
A3: card Coim(h^f,x) = card Coim(f,x)+card(h"{x}) & card((h^g)"{x}) =
    card(g"{x} )+card(h"{x}) by FINSEQ_3:57;
    card Coim(h^f,x) = card Coim(h^g,x) by A2;
    hence card Coim(f,x) = card Coim(g,x) by A3;
  end;
  hence thesis;
end;
