reserve I for set;
reserve S for non empty non void ManySortedSign,
  U0, U1 for non-empty MSAlgebra over S;
reserve s for SortSymbol of S;
reserve e for Element of (Equations S).s;
reserve E for EqualSet of S;

theorem Th37:
  for F being MSAlgebra-Family of I, S st (for i being set st i in
  I ex A being MSAlgebra over S st A = F.i & A |= e) holds product F |= e
proof
  let F be MSAlgebra-Family of I, S such that
A1: for i being set st i in I ex A being MSAlgebra over S st A = F.i & A |= e;
  per cases;
  suppose
    I = {};
    then reconsider J = I as empty set;
    reconsider FJ = F as MSAlgebra-Family of J, S;
    thus product F |= e
    proof
      let h be ManySortedFunction of TermAlg S, product F such that
      h is_homomorphism TermAlg S, product F;
A2:   (the Sorts of product FJ).s = product Carrier(FJ,s) by PRALG_2:def 10
        .= {{}} by CARD_3:10;
A3:   h.s.(e`2) in (the Sorts of product FJ).s by Th30,FUNCT_2:5;
      h.s.(e`1) in (the Sorts of product FJ).s by Th29,FUNCT_2:5;
      hence h.s.(e`1) = {} by A2,TARSKI:def 1
        .= h.s.(e`2) by A2,A3,TARSKI:def 1;
    end;
  end;
  suppose
    I <> {};
    then reconsider J = I as non empty set;
    reconsider F1 = F as MSAlgebra-Family of J, S;
    now
      let i be Element of J;
      ex A being MSAlgebra over S st A = F1.i & A |= e by A1;
      hence F1.i |= e;
    end;
    hence thesis by Lm1;
  end;
end;
