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 Th38:
  @p <> 'not' A & @p <> (#)A & @p <> A '&' B
proof
  set e2 = elementary_tree 2;
  set e1 = elementary_tree 1;
  set e0 = elementary_tree 0;
A1: dom @p = e0 by FUNCOP_1:13;
A2: <*0*> in e1 by TARSKI:def 2,TREES_1:51;
  dom (e1 --> [1,0]) = e1 by FUNCOP_1:13;
  then dom ('not' A) = e1 with-replacement (<*0*>,dom A) by A2,TREES_2:def 11;
  then <*0*> in dom ('not' A) by A2,TREES_1:def 9;
  hence @p <> 'not' A by A1,TARSKI:def 1,TREES_1:29;
  dom (e1 --> [1,1]) = e1 by FUNCOP_1:13;
  then dom ((#)A) = e1 with-replacement (<*0*>,dom A) by A2,TREES_2:def 11;
  then <*0*> in dom ((#)A) by A2,TREES_1:def 9;
  hence @p <> (#)A by A1,TARSKI:def 1,TREES_1:29;
  set y = (e2-->[2,0]) with-replacement (<*0*>,A);
A3: <*1*> in e2 & not <*0*> is_a_proper_prefix_of <*1*> by TREES_1:28,52;
A4: <*0*> in e2 & dom (e2 --> [2,0]) = e2 by FUNCOP_1:13,TREES_1:28;
  then dom y = dom(e2-->[2,0]) with-replacement (<*0*>,dom A) by TREES_2:def 11
;
  then
A5: <*1*> in dom y by A4,A3,TREES_1:def 9;
  then dom (A '&' B) = dom y with-replacement (<*1*>,dom B) by TREES_2:def 11;
  then <*1*> in dom (A '&' B) by A5,TREES_1:def 9;
  hence thesis by A1,TARSKI:def 1,TREES_1:29;
end;
