let A be QC-alphabet ; :: thesis: for F being Element of QC-WFF A
for G1, G2 being Subformula of F holds { (t ^ s) where t is Entry_Point_in_Subformula_Tree of G1, s is Element of dom (tree_of_subformulae G1) : s in G1 -entry_points_in_subformula_tree_of G2 } = { (t ^ s) where t is Element of dom (tree_of_subformulae F), s is Element of dom (tree_of_subformulae G1) : ( t in F -entry_points_in_subformula_tree_of G1 & s in G1 -entry_points_in_subformula_tree_of G2 ) }
let F be Element of QC-WFF A; :: thesis: for G1, G2 being Subformula of F holds { (t ^ s) where t is Entry_Point_in_Subformula_Tree of G1, s is Element of dom (tree_of_subformulae G1) : s in G1 -entry_points_in_subformula_tree_of G2 } = { (t ^ s) where t is Element of dom (tree_of_subformulae F), s is Element of dom (tree_of_subformulae G1) : ( t in F -entry_points_in_subformula_tree_of G1 & s in G1 -entry_points_in_subformula_tree_of G2 ) }
let G1, G2 be Subformula of F; :: thesis: { (t ^ s) where t is Entry_Point_in_Subformula_Tree of G1, s is Element of dom (tree_of_subformulae G1) : s in G1 -entry_points_in_subformula_tree_of G2 } = { (t ^ s) where t is Element of dom (tree_of_subformulae F), s is Element of dom (tree_of_subformulae G1) : ( t in F -entry_points_in_subformula_tree_of G1 & s in G1 -entry_points_in_subformula_tree_of G2 ) }
thus { (t ^ s) where t is Entry_Point_in_Subformula_Tree of G1, s is Element of dom (tree_of_subformulae G1) : s in G1 -entry_points_in_subformula_tree_of G2 } c= { (t ^ s) where t is Element of dom (tree_of_subformulae F), s is Element of dom (tree_of_subformulae G1) : ( t in F -entry_points_in_subformula_tree_of G1 & s in G1 -entry_points_in_subformula_tree_of G2 ) } :: according to XBOOLE_0:def 10 :: thesis: { (t ^ s) where t is Element of dom (tree_of_subformulae F), s is Element of dom (tree_of_subformulae G1) : ( t in F -entry_points_in_subformula_tree_of G1 & s in G1 -entry_points_in_subformula_tree_of G2 ) } c= { (t ^ s) where t is Entry_Point_in_Subformula_Tree of G1, s is Element of dom (tree_of_subformulae G1) : s in G1 -entry_points_in_subformula_tree_of G2 }

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { (t ^ s) where t is Entry_Point_in_Subformula_Tree of G1, s is Element of dom (tree_of_subformulae G1) : s in G1 -entry_points_in_subformula_tree_of G2 } or x in { (t ^ s) where t is Element of dom (tree_of_subformulae F), s is Element of dom (tree_of_subformulae G1) : ( t in F -entry_points_in_subformula_tree_of G1 & s in G1 -entry_points_in_subformula_tree_of G2 ) } )
assume x in { (t ^ s) where t is Entry_Point_in_Subformula_Tree of G1, s is Element of dom (tree_of_subformulae G1) : s in G1 -entry_points_in_subformula_tree_of G2 } ; :: thesis: x in { (t ^ s) where t is Element of dom (tree_of_subformulae F), s is Element of dom (tree_of_subformulae G1) : ( t in F -entry_points_in_subformula_tree_of G1 & s in G1 -entry_points_in_subformula_tree_of G2 ) }
then consider t1 being Entry_Point_in_Subformula_Tree of G1, s1 being Element of dom (tree_of_subformulae G1) such that
A1: ( x = t1 ^ s1 & s1 in G1 -entry_points_in_subformula_tree_of G2 ) ;
(tree_of_subformulae F) . t1 = G1 by Def5;
then t1 in F -entry_points_in_subformula_tree_of G1 by Def3;
hence x in { (t ^ s) where t is Element of dom (tree_of_subformulae F), s is Element of dom (tree_of_subformulae G1) : ( t in F -entry_points_in_subformula_tree_of G1 & s in G1 -entry_points_in_subformula_tree_of G2 ) } by A1; :: thesis: verum

end;

thus { (t ^ s) where t is Element of dom (tree_of_subformulae F), s is Element of dom (tree_of_subformulae G1) : ( t in F -entry_points_in_subformula_tree_of G1 & s in G1 -entry_points_in_subformula_tree_of G2 ) } c= { (t ^ s) where t is Entry_Point_in_Subformula_Tree of G1, s is Element of dom (tree_of_subformulae G1) : s in G1 -entry_points_in_subformula_tree_of G2 } :: thesis: verum

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { (t ^ s) where t is Element of dom (tree_of_subformulae F), s is Element of dom (tree_of_subformulae G1) : ( t in F -entry_points_in_subformula_tree_of G1 & s in G1 -entry_points_in_subformula_tree_of G2 ) } or x in { (t ^ s) where t is Entry_Point_in_Subformula_Tree of G1, s is Element of dom (tree_of_subformulae G1) : s in G1 -entry_points_in_subformula_tree_of G2 } )
assume x in { (t ^ s) where t is Element of dom (tree_of_subformulae F), s is Element of dom (tree_of_subformulae G1) : ( t in F -entry_points_in_subformula_tree_of G1 & s in G1 -entry_points_in_subformula_tree_of G2 ) } ; :: thesis: x in { (t ^ s) where t is Entry_Point_in_Subformula_Tree of G1, s is Element of dom (tree_of_subformulae G1) : s in G1 -entry_points_in_subformula_tree_of G2 }
then consider t1 being Element of dom (tree_of_subformulae F), s1 being Element of dom (tree_of_subformulae G1) such that
A2: x = t1 ^ s1 and
A3: t1 in F -entry_points_in_subformula_tree_of G1 and
A4: s1 in G1 -entry_points_in_subformula_tree_of G2 ;
(tree_of_subformulae F) . t1 = G1 by A3, Def3;
then reconsider t1 = t1 as Entry_Point_in_Subformula_Tree of G1 by Def5;
x = t1 ^ s1 by A2;
hence x in { (t ^ s) where t is Entry_Point_in_Subformula_Tree of G1, s is Element of dom (tree_of_subformulae G1) : s in G1 -entry_points_in_subformula_tree_of G2 } by A4; :: thesis: verum

end;