reserve a, I for set,
  S for non empty non void ManySortedSign;

theorem
  for I being non empty set for s being Element of S for A being
MSAlgebra-Family of I,S for f, g being Element of product Carrier(A,s) st for a
being Element of I holds proj(Carrier(A,s),a).f = proj(Carrier(A,s),a).g holds
  f = g
proof
  let I be non empty set, s be Element of S, A be MSAlgebra-Family of I,S, f,
  g be Element of product Carrier(A,s) such that
A1: for a being Element of I holds proj(Carrier(A,s),a).f = proj(Carrier
  (A,s),a).g;
  now
 dom f = dom Carrier(A,s) by CARD_3:9;
    hence dom f = dom g by CARD_3:9;
    let x be object such that
A2: x in dom f;
A3: dom (proj(Carrier(A,s),x)) = product Carrier(A,s) by CARD_3:def 16;
    hence f.x = proj(Carrier(A,s),x).f by CARD_3:def 16
      .= proj(Carrier(A,s),x).g by A1,A2
      .= g.x by A3,CARD_3:def 16;
  end;
  hence thesis;
end;
