let A, B be MP-wff; :: thesis:
set e2 = elementary_tree 2;
set e1 = elementary_tree 1;
A1: dom VERUM = elementary_tree 0 by FUNCOP_1:19;
set F = (elementary_tree 1) --> [1,1];
set E = (elementary_tree 1) --> [1,0 ];
A2: <*0 *> in elementary_tree 1 by TARSKI:def 2, TREES_1:88;
then <*0 *> in dom ((elementary_tree 1) --> [1,0 ]) by FUNCOP_1:19;
then ( dom ((elementary_tree 1) --> [1,0 ]) = elementary_tree 1 & dom ('not' A) = (dom ((elementary_tree 1) --> [1,0 ])) with-replacement <*0 *>,(dom A) ) by FUNCOP_1:19, TREES_2:def 12;
then <*0 *> in dom ('not' A) by A2, TREES_1:def 12;
hence VERUM <> 'not' A by A1, TARSKI:def 1, TREES_1:56; :: thesis:
<*0 *> in dom ((elementary_tree 1) --> [1,1]) by A2, FUNCOP_1:19;
then ( dom ((elementary_tree 1) --> [1,1]) = elementary_tree 1 & dom ((#) A) = (dom ((elementary_tree 1) --> [1,1])) with-replacement <*0 *>,(dom A) ) by FUNCOP_1:19, TREES_2:def 12;
then <*0 *> in dom ((#) A) by A2, TREES_1:def 12;
hence VERUM <> (#) A by A1, TARSKI:def 1, TREES_1:56; :: thesis:
set y = ((elementary_tree 2) --> [2,0 ]) with-replacement <*0 *>,A;
A3: ( <*1*> in elementary_tree 2 & not <*0 *> is_a_proper_prefix_of <*1*> ) by TREES_1:55, TREES_1:89;
A4: ( <*0 *> in elementary_tree 2 & dom ((elementary_tree 2) --> [2,0 ]) = elementary_tree 2 ) by FUNCOP_1:19, TREES_1:55;
then dom (((elementary_tree 2) --> [2,0 ]) with-replacement <*0 *>,A) = (dom ((elementary_tree 2) --> [2,0 ])) with-replacement <*0 *>,(dom A) by TREES_2:def 12;
then A5: <*1*> in dom (((elementary_tree 2) --> [2,0 ]) with-replacement <*0 *>,A) by A4, A3, TREES_1:def 12;
then dom (A '&' B) = (dom (((elementary_tree 2) --> [2,0 ]) with-replacement <*0 *>,A)) with-replacement <*1*>,(dom B) by TREES_2:def 12;
then A6: <*1*> in dom (A '&' B) by A5, TREES_1:def 12;
assume not VERUM <> A '&' B ; :: thesis:
hence contradiction by A1, A6, TARSKI:def 1, TREES_1:56; :: thesis: