reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;

theorem
  for c being Element of Constructors holds
  (kind_of c = a_Type iff c in Modes) &
  (kind_of c = an_Adj iff c in Attrs) &
  (kind_of c = a_Term iff c in Funcs)
proof
  let x be Element of Constructors;
A1: x in Modes \/ Attrs or x in Funcs by XBOOLE_0:def 3;
A2: x in Modes implies x`1 in {a_Type} by MCART_1:10;
A3: x in Attrs implies x`1 in {an_Adj} by MCART_1:10;
  x in Funcs implies x`1 in {a_Term} by MCART_1:10;
  hence thesis by A1,A2,A3,TARSKI:def 1,XBOOLE_0:def 3;
end;
