let F, G, H be ZF-formula; ( not F is_proper_subformula_of G '&' H or F is_subformula_of G or F is_subformula_of H )
assume that
A1:
F is_subformula_of G '&' H
and
A2:
F <> G '&' H
; ZF_LANG:def 41 ( F is_subformula_of G or 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 = G '&' 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;
(1 + 1) + k = (1 + k) + 1
;
then
1 + k < n
by A8, NAT_1:13;
then consider H1, G1 being ZF-formula such that
A9:
L . (1 + k) = H1
and
A10:
( L . ((1 + k) + 1) = G1 & H1 is_immediate_constituent_of G1 )
by A7, NAT_1:11;
reconsider L1 = L | (Seg (1 + k)) as FinSequence by FINSEQ_1:15;
F is_subformula_of H1
proof
take m = 1
+ k;
ZF_LANG:def 40 ex L being FinSequence st
( 1 <= m & len L = m & L . 1 = F & L . m = H1 & ( 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
;
( 1 <= m & len L1 = m & L1 . 1 = F & L1 . m = H1 & ( 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 A11:
1
<= m
by NAT_1:11;
( len L1 = m & L1 . 1 = F & L1 . m = H1 & ( 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;
( L1 . 1 = F & L1 . m = H1 & ( 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 ) ) )
A12:
now for j being Nat st 1 <= j & j <= m holds
L1 . j = L . jend;
hence
L1 . 1
= F
by A5, A11;
( L1 . m = H1 & ( 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
L1 . m = H1
by A9, A11, A12;
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 ;
( 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
;
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
;
ex F1 being ZF-formula st
( L1 . j = F1 & L1 . (j + 1) = F1 & F1 is_immediate_constituent_of F1 )
take
G1
;
( 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 A12, A13, A14, A15;
verum
end;
hence
( F is_subformula_of G or F is_subformula_of H )
by A6, A8, A10, Th53; verum