reserve i,j for Nat;

theorem
 for C being initialized standardized ConstructorSignature
 for a being quasi-adjective of C holds
   a is negative iff (a .{})`1 = non_op
  proof let C be initialized standardized ConstructorSignature;
   let t be quasi-adjective of C;
   per cases;
   suppose
A1:  t is positive expression of C, an_Adj C; then
     (t.{})`1 in Constructors & non_op in {*, non_op} by Th24,TARSKI:def 2;
    hence thesis by A1,ABCMIZ_1:39,XBOOLE_0:3;
   end;
   suppose
A2:  t is negative expression of C, an_Adj C; then
    consider a being expression of C, an_Adj C such that
A3:  a is positive & t = (non_op C)term a by ABCMIZ_1:def 38;
     t = [non_op, the carrier of C]-tree <*a*> by A3,ABCMIZ_1:43; then
     t.{} = [non_op, the carrier of C] by TREES_4:def 4;
    hence thesis by A2;
   end;
  end;
