let A, B, C be MP-wff; :: thesis: ( 'not' A <> (#) B & 'not' A <> B '&' C )
set e2 = elementary_tree 2;
set e1 = elementary_tree 1;
set E = (elementary_tree 1) --> [1,0 ];
set F = (elementary_tree 1) --> [1,1];
reconsider e = {} as Element of dom ('not' A) by TREES_1:47;
A1: ( {} in dom ((#) B) & ( for u being FinSequence of NAT holds
( not u in dom B or not e = <*0 *> ^ u or not ((#) B) . e = B . u ) ) ) by TREES_1:47;
A2: <*0 *> in elementary_tree 1 by TARSKI:def 2, TREES_1:88;
then A3: <*0 *> in dom ((elementary_tree 1) --> [1,0 ]) by FUNCOP_1:19;
then A4: dom ('not' A) = (dom ((elementary_tree 1) --> [1,0 ])) with-replacement <*0 *>,(dom A) by TREES_2:def 12;
A5: <*0 *> in dom ((elementary_tree 1) --> [1,1]) by A2, FUNCOP_1:19;
then dom ((#) B) = (dom ((elementary_tree 1) --> [1,1])) with-replacement <*0 *>,(dom B) by TREES_2:def 12;
then A6: ((#) B) . e = ((elementary_tree 1) --> [1,1]) . e by A5, A1, TREES_2:def 12;
e in elementary_tree 1 by TREES_1:47;
then A7: ( ((elementary_tree 1) --> [1,0 ]) . e = [1,0 ] & ((elementary_tree 1) --> [1,1]) . e = [1,1] ) by FUNCOP_1:13;
( ( for u being FinSequence of NAT holds
( not u in dom A or not e = <*0 *> ^ u or not ('not' A) . e = A . u ) ) & [1,0 ] <> [1,1] ) by ZFMISC_1:33;
hence 'not' A <> (#) B by A3, A4, A6, A7, TREES_2:def 12; :: thesis: 'not' A <> B '&' C
set y = ((elementary_tree 2) --> [2,0 ]) with-replacement <*0 *>,B;
A8: ( <*1*> in elementary_tree 2 & not <*0 *> is_a_proper_prefix_of <*1*> ) by TREES_1:55, TREES_1:89;
A9: ( <*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 *>,B) = (dom ((elementary_tree 2) --> [2,0 ])) with-replacement <*0 *>,(dom B) by TREES_2:def 12;
then A10: <*1*> in dom (((elementary_tree 2) --> [2,0 ]) with-replacement <*0 *>,B) by A9, A8, TREES_1:def 12;
then dom (B '&' C) = (dom (((elementary_tree 2) --> [2,0 ]) with-replacement <*0 *>,B)) with-replacement <*1*>,(dom C) by TREES_2:def 12;
then A11: <*1*> in dom (B '&' C) by A10, TREES_1:def 12;
A12: dom ((elementary_tree 1) --> [1,0 ]) = elementary_tree 1 by FUNCOP_1:19;

A13: now

assume <*1*> in dom ((elementary_tree 1) --> [1,0 ]) ; :: thesis: contradiction
then ( <*1*> = {} or <*1*> = <*0 *> ) by A12, TARSKI:def 2, TREES_1:88;
hence contradiction by TREES_1:16; :: thesis: verum

end;

assume not 'not' A <> B '&' C ; :: thesis: contradiction
then ex s being FinSequence of NAT st
( s in dom A & <*1*> = <*0 *> ^ s ) by A3, A4, A11, A13, TREES_1:def 12;
then <*0 *> is_a_prefix_of <*1*> by TREES_1:8;
hence contradiction by TREES_1:16; :: thesis: verum