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 Th40:
  (#)A <> B '&' C
proof
  set e2 = elementary_tree 2;
  set e1 = elementary_tree 1;
  set F = e1 --> [1,1];
  set y = (e2-->[2,0]) with-replacement (<*0*>,B);
A1: <*1*> in e2 & not <*0*> is_a_proper_prefix_of <*1*> by TREES_1:28,52;
A2: <*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 B) by TREES_2:def 11
;
  then
A3: <*1*> in dom y by A2,A1,TREES_1:def 9;
  then dom (B '&' C) = dom y with-replacement (<*1*>,dom C) by TREES_2:def 11;
  then
A4: <*1*> in dom (B '&' C) by A3,TREES_1:def 9;
  assume
A5: not thesis;
A6: now
    assume <*1*> in dom F;
    then <*1*> = {} or <*1*> = <*0*> by TARSKI:def 2,TREES_1:51;
    hence contradiction by TREES_1:3;
  end;
  <*0*> in e1 by TARSKI:def 2,TREES_1:51;
  then
A7: <*0*> in dom F by FUNCOP_1:13;
  then dom (#)A = dom F with-replacement (<*0*>,dom A) by TREES_2:def 11;
  then ex s st s in dom A & <*1*> = <*0*>^s by A7,A4,A6,A5,TREES_1:def 9;
  then <*0*> is_a_prefix_of <*1*> by TREES_1:1;
  hence contradiction by TREES_1:3;
end;
