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

theorem Th1:
  for f,g,h be FinSequence holds
  f,g are_fiberwise_equipotent iff f^h, g^h are_fiberwise_equipotent
proof
  let f,g,h be FinSequence;
  thus f,g are_fiberwise_equipotent implies f^h, g^h 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(f^h,y) = card Coim(g,y) + card Coim(h,y) by FINSEQ_3:57
        .= card Coim(g^h,y) by FINSEQ_3:57;
    end;
    hence thesis;
  end;
  assume
A2: f^h,g^h are_fiberwise_equipotent;
  now
    let x be object;
A3: card Coim(f^h,x) = card Coim(f,x)+card Coim(h,x) & card Coim(g^h,x) =
    card Coim(g,x)+card Coim(h,x) by FINSEQ_3:57;
    card Coim(f^h,x) = card Coim(g^h,x) by A2;
    hence card Coim(f,x) = card Coim(g,x) by A3;
  end;
  hence thesis;
end;
