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
  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;
  let s be SortSymbol of S, e be Element of (Equations S).s such that
A2: e in E.s;
  now
    let i be set;
    assume i in I;
    then consider A being MSAlgebra over S such that
A3: A = F.i & A |= E by A1;
    take A;
    thus A = F.i & A |= e by A2,A3;
  end;
  hence thesis by Th37;
end;
