let A, B, C be MP-wff; (#) A <> B '&' C
set e2 = elementary_tree 2;
set e1 = elementary_tree 1;
set F = (elementary_tree 1) --> [1,1];
set y = ((elementary_tree 2) --> [2,0]) with-replacement (<*0*>,B);
A1:
( <*1*> in elementary_tree 2 & not <*0*> is_a_proper_prefix_of <*1*> )
by TREES_1:28, TREES_1:52;
A2:
( <*0*> in elementary_tree 2 & dom ((elementary_tree 2) --> [2,0]) = elementary_tree 2 )
by FUNCOP_1:13, TREES_1:28;
then
dom (((elementary_tree 2) --> [2,0]) with-replacement (<*0*>,B)) = (dom ((elementary_tree 2) --> [2,0])) with-replacement (<*0*>,(dom B))
by TREES_2:def 11;
then A3:
<*1*> in dom (((elementary_tree 2) --> [2,0]) with-replacement (<*0*>,B))
by A2, A1, TREES_1:def 9;
then
dom (B '&' C) = (dom (((elementary_tree 2) --> [2,0]) with-replacement (<*0*>,B))) 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 (#) A <> B '&' C
; contradiction
<*0*> in elementary_tree 1
by TARSKI:def 2, TREES_1:51;
then A7:
<*0*> in dom ((elementary_tree 1) --> [1,1])
by FUNCOP_1:13;
then
dom ((#) A) = (dom ((elementary_tree 1) --> [1,1])) with-replacement (<*0*>,(dom A))
by TREES_2:def 11;
then
ex s being FinSequence of NAT 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; verum