theorem Th15:
  for o being OperSymbol of S st
  o = In((the connectives of S).5, the carrier' of S)
  holds the_arity_of o = {} & the_result_sort_of o = I
  proof
    let o be OperSymbol of S;
    assume A1: o = In((the connectives of S).5, the carrier' of S);
    4+6 <= len the connectives of S by AOFA_A00:def 39;
    then 5 <= len the connectives of S by XXREAL_0:2;
    then 5 in dom the connectives of S by FINSEQ_3:25;
    then o = (the connectives of S).5 by A1,FUNCT_1:102,SUBSET_1:def 8;
    then o is_of_type {}, I by AOFA_A00:53;
    hence the_arity_of o = {} & the_result_sort_of o = I;
  end;
