reserve x,y,x1,x2,z for set,
  n,m,k for Nat,
  t1 for (DecoratedTree of [: NAT,NAT :]),
  w,s,t,u for FinSequence of NAT,
  D for non empty set;
reserve s9,w9,v9 for Element of NAT*;
reserve p,q for MP-variable;
reserve A,A1,B,B1,C,C1 for MP-wff;

theorem
  for A being Element of MP-WFF holds A = VERUM or A is atomic MP-wff or
  A is negative MP-wff or A is necessitive MP-wff or A is conjunctive MP-wff
proof
  defpred Prop[Element of MP-WFF] means $1 = VERUM or $1 is atomic MP-wff or
$1 is negative MP-wff or $1 is necessitive MP-wff or $1 is conjunctive MP-wff;
A1: Prop[VERUM];
A2: for A being Element of MP-WFF st Prop[A] holds Prop['not' A] by Def17;
A3: for A,B being Element of MP-WFF st Prop[A] & Prop[B] holds Prop[A '&' B]
  by Def19;
A4: for A being Element of MP-WFF st Prop[A] holds Prop[(#) A] by Def18;
A5: for p being MP-variable holds Prop[@p] by Def16;
  thus for A be Element of MP-WFF holds Prop[A] from MPInd(A1,A5, A2,A4,A3);
end;
