let F, G be LTL-formula; :: thesis:
assume that
A1: ( G is conjunctive or G is disjunctive or G is Until or G is Release ) and
A2: F is_subformula_of G and
A3: F <> G ; :: according to MODELC_2:def 23 :: thesis:
consider n being Nat, L being FinSequence such that
A4: 1 <= n and
A5: len L = n and
A6: L . 1 = F and
A7: L . n = G and
A8: for k being Nat st 1 <= k & k < n holds
ex H1, F1 being LTL-formula st
( L . k = H1 & L . (k + 1) = F1 & H1 is_immediate_constituent_of F1 ) by A2;
1 < n by A3, A4, A6, A7, XXREAL_0:1;
then 1 + 1 <= n by NAT_1:13;
then consider k being Nat such that
A9: n = 2 + k by NAT_1:10;
reconsider L1 = L | (Seg (1 + k)) as FinSequence by FINSEQ_1:15;
(1 + 1) + k = (1 + k) + 1 ;
then 1 + k < n by A9, NAT_1:13;
then consider H1, G1 being LTL-formula such that
A10: L . (1 + k) = H1 and
A11: ( L . ((1 + k) + 1) = G1 & H1 is_immediate_constituent_of G1 ) by A8, NAT_1:11;
F is_subformula_of H1

let j be Nat; :: thesis:
assume ( 1 <= j & j <= m ) ; :: thesis:
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:

end;

hence L1 . 1 = F by A6, A12; :: thesis:
thus L1 . m = H1 by A10, A12, A13; :: thesis:
let j be Nat; :: thesis:
assume that
A14: 1 <= j and
A15: j < m ; :: thesis:
m <= m + 1 by NAT_1:11;
then j < n by A9, A15, XXREAL_0:2;
then consider F1, G1 being LTL-formula such that
A16: ( L . j = F1 & L . (j + 1) = G1 & F1 is_immediate_constituent_of G1 ) by A8, A14;
take F1 ; :: thesis:
take G1 ; :: thesis:
( 1 <= 1 + j & j + 1 <= m ) by A14, A15, NAT_1:13;
hence ( L1 . j = F1 & L1 . (j + 1) = G1 & F1 is_immediate_constituent_of G1 ) by A13, A14, A15, A16; :: thesis:

end;