let A be QC-alphabet ; :: thesis:
let F, H be Element of QC-WFF A; :: thesis:
thus Subformulae (H '&' F) c= ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)} :: according to XBOOLE_0:def 10 :: thesis:

proof

let a be object ; :: according to TARSKI:def 3 :: thesis:
assume a in Subformulae (H '&' F) ; :: thesis:
then consider G being Element of QC-WFF A such that
A1: G = a and
A2: G is_subformula_of H '&' F by Def22;

now :: thesis:

assume G <> H '&' F ; :: thesis:
then G is_proper_subformula_of H '&' F by A2;
then ( G is_subformula_of H or G is_subformula_of F ) by Th69;
then ( a in Subformulae H or a in Subformulae F ) by A1, Def22;
hence a in (Subformulae H) \/ (Subformulae F) by XBOOLE_0:def 3; :: thesis:

end;

then ( a in (Subformulae H) \/ (Subformulae F) or a in {(H '&' F)} ) by A1, TARSKI:def 1;
hence a in ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)} by XBOOLE_0:def 3; :: thesis:

end;

let a be object ; :: according to TARSKI:def 3 :: thesis:
assume A3: a in ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)} ; :: thesis:

A4: now :: thesis:

assume a in {(H '&' F)} ; :: thesis:
then a = H '&' F by TARSKI:def 1;
hence a in Subformulae (H '&' F) by Def22; :: thesis:

end;

A5: now :: thesis:

assume a in Subformulae F ; :: thesis:
then consider G being Element of QC-WFF A such that
A6: G = a and
A7: G is_subformula_of F by Def22;
F is_immediate_constituent_of H '&' F ;
then F is_proper_subformula_of H '&' F by Th53;
then F is_subformula_of H '&' F ;
then G is_subformula_of H '&' F by A7, Th57;
hence a in Subformulae (H '&' F) by A6, Def22; :: thesis:

end;

A8: now :: thesis:

assume a in Subformulae H ; :: thesis:
then consider G being Element of QC-WFF A such that
A9: G = a and
A10: G is_subformula_of H by Def22;
H is_immediate_constituent_of H '&' F ;
then H is_proper_subformula_of H '&' F by Th53;
then H is_subformula_of H '&' F ;
then G is_subformula_of H '&' F by A10, Th57;
hence a in Subformulae (H '&' F) by A9, Def22; :: thesis:

end;

( not a in (Subformulae H) \/ (Subformulae F) or a in Subformulae H or a in Subformulae F ) by XBOOLE_0:def 3;
hence a in Subformulae (H '&' F) by A3, A8, A5, A4, XBOOLE_0:def 3; :: thesis: