let F, H be ZF-formula; :: thesis: ( F is_proper_subformula_of 'not' H implies F is_subformula_of H )
assume that
A1: F is_subformula_of 'not' H and
A2: F <> 'not' H ; :: according to ZF_LANG:def 41 :: thesis: F is_subformula_of H
consider n being Element of NAT , L being FinSequence such that
A3: 1 <= n and
A4: len L = n and
A5: L . 1 = F and
A6: L . n = 'not' H and
A7: for k being Element of NAT st 1 <= k & k < n holds
ex H1, F1 being ZF-formula st
( L . k = H1 & L . (k + 1) = F1 & H1 is_immediate_constituent_of F1 ) by A1;
1 < n by A2, A3, A5, A6, XXREAL_0:1;
then 1 + 1 <= n by NAT_1:13;
then consider k being Nat such that
A8: n = 2 + k by NAT_1:10;
reconsider k = k as Element of NAT by ORDINAL1:def 12;
reconsider L1 = L | (Seg (1 + k)) as FinSequence by FINSEQ_1:15;
take m = 1 + k; :: according to ZF_LANG:def 40 :: thesis: ex L being FinSequence st
( 1 <= m & len L = m & L . 1 = F & L . m = H & ( for k being Element of NAT st 1 <= k & k < m holds
ex H1, F1 being ZF-formula st
( L . k = H1 & L . (k + 1) = F1 & H1 is_immediate_constituent_of F1 ) ) )
take L1 ; :: thesis: ( 1 <= m & len L1 = m & L1 . 1 = F & L1 . m = H & ( for k being Element of NAT st 1 <= k & k < m holds
ex H1, F1 being ZF-formula st
( L1 . k = H1 & L1 . (k + 1) = F1 & H1 is_immediate_constituent_of F1 ) ) )
thus A9: 1 <= m by NAT_1:11; :: thesis: ( len L1 = m & L1 . 1 = F & L1 . m = H & ( for k being Element of NAT st 1 <= k & k < m holds
ex H1, F1 being ZF-formula st
( L1 . k = H1 & L1 . (k + 1) = F1 & H1 is_immediate_constituent_of F1 ) ) )
1 + k <= (1 + k) + 1 by NAT_1:11;
hence len L1 = m by A4, A8, FINSEQ_1:17; :: thesis: ( L1 . 1 = F & L1 . m = H & ( for k being Element of NAT st 1 <= k & k < m holds
ex H1, F1 being ZF-formula st
( L1 . k = H1 & L1 . (k + 1) = F1 & H1 is_immediate_constituent_of F1 ) ) )

A10: now :: thesis: for j being Nat st 1 <= j & j <= m holds
L1 . j = L . j

let j be Nat; :: thesis: ( 1 <= j & j <= m implies L1 . j = L . j )
assume ( 1 <= j & j <= m ) ; :: thesis: L1 . j = L . j
then j in { j1 where j1 is Nat : ( 1 <= j1 & j1 <= 1 + k ) } ;
then j in Seg (1 + k) by FINSEQ_1:def 1;
hence L1 . j = L . j by FUNCT_1:49; :: thesis: verum

end;

hence L1 . 1 = F by A5, A9; :: thesis: ( L1 . m = H & ( for k being Element of NAT st 1 <= k & k < m holds
ex H1, F1 being ZF-formula st
( L1 . k = H1 & L1 . (k + 1) = F1 & H1 is_immediate_constituent_of F1 ) ) )
m < m + 1 by NAT_1:13;
then consider F1, G1 being ZF-formula such that
A11: L . m = F1 and
A12: ( L . (m + 1) = G1 & F1 is_immediate_constituent_of G1 ) by A7, A8, NAT_1:11;
F1 = H by A6, A8, A12, Th52;
hence L1 . m = H by A9, A10, A11; :: thesis: for k being Element of NAT st 1 <= k & k < m holds
ex H1, F1 being ZF-formula st
( L1 . k = H1 & L1 . (k + 1) = F1 & H1 is_immediate_constituent_of F1 )
let j be Element of NAT ; :: thesis: ( 1 <= j & j < m implies ex H1, F1 being ZF-formula st
( L1 . j = H1 & L1 . (j + 1) = F1 & H1 is_immediate_constituent_of F1 ) )
assume that
A13: 1 <= j and
A14: j < m ; :: thesis: ex H1, F1 being ZF-formula st
( L1 . j = H1 & L1 . (j + 1) = F1 & H1 is_immediate_constituent_of F1 )
m <= m + 1 by NAT_1:11;
then j < n by A8, A14, XXREAL_0:2;
then consider F1, G1 being ZF-formula such that
A15: ( L . j = F1 & L . (j + 1) = G1 & F1 is_immediate_constituent_of G1 ) by A7, A13;
take F1 ; :: thesis: ex F1 being ZF-formula st
( L1 . j = F1 & L1 . (j + 1) = F1 & F1 is_immediate_constituent_of F1 )
take G1 ; :: thesis: ( L1 . j = F1 & L1 . (j + 1) = G1 & F1 is_immediate_constituent_of G1 )
( 1 <= 1 + j & j + 1 <= m ) by A13, A14, NAT_1:13;
hence ( L1 . j = F1 & L1 . (j + 1) = G1 & F1 is_immediate_constituent_of G1 ) by A10, A13, A14, A15; :: thesis: verum