theorem Th12:
  for o being OperSymbol of S st
  o = In((the connectives of S).2, the carrier' of S)
  holds o = (the connectives of S).2 &
  the_arity_of o = <*the bool-sort of S*> &
  the_result_sort_of o = the bool-sort of S
  proof
    let o be OperSymbol of S;
    assume A1: o = In((the connectives of S).2, the carrier' of S);
    4+6 <= len the connectives of S by AOFA_A00:def 39;
    then 2 <= len the connectives of S by XXREAL_0:2;
    then 2 in dom the connectives of S by FINSEQ_3:25;
    hence o = (the connectives of S).2 by A1,FUNCT_1:102,SUBSET_1:def 8;
    then o is_of_type <*the bool-sort of S*>, the bool-sort of S
    by AOFA_A00:def 31;
    hence the_arity_of o = <*the bool-sort of S*> &
    the_result_sort_of o = the bool-sort of S;
  end;
