let F be Element of QC-WFF ; :: thesis: for t, t' being Element of dom (tree_of_subformulae F) st t is_a_proper_prefix_of t' holds
(tree_of_subformulae F) . t' is_proper_subformula_of (tree_of_subformulae F) . t
let t, t' be Element of dom (tree_of_subformulae F); :: thesis: ( t is_a_proper_prefix_of t' implies (tree_of_subformulae F) . t' is_proper_subformula_of (tree_of_subformulae F) . t )
set G = (tree_of_subformulae F) . t;
set H = (tree_of_subformulae F) . t';
assume A1: t is_a_proper_prefix_of t' ; :: thesis: (tree_of_subformulae F) . t' is_proper_subformula_of (tree_of_subformulae F) . t
then t is_a_prefix_of t' by XBOOLE_0:def 8;
then A2: (tree_of_subformulae F) . t' is_subformula_of (tree_of_subformulae F) . t by Th42;
(tree_of_subformulae F) . t' <> (tree_of_subformulae F) . t by A1, Th44;
hence (tree_of_subformulae F) . t' is_proper_subformula_of (tree_of_subformulae F) . t by A2, QC_LANG2:def 22; :: thesis: verum