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

proof

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in Subformulae (H '&' F) or a in ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)} )
assume a in Subformulae (H '&' F) ; :: thesis: a in ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)}
then consider G being Element of QC-WFF such that
A1: ( G = a & G is_subformula_of H '&' F ) by Def23;

now

assume G <> H '&' F ; :: thesis: a in (Subformulae H) \/ (Subformulae F)
then G is_proper_subformula_of H '&' F by A1, Def22;
then ( G is_subformula_of H or G is_subformula_of F ) by Th89;
then ( a in Subformulae H or a in Subformulae F ) by A1, Def23;
hence a in (Subformulae H) \/ (Subformulae F) by XBOOLE_0:def 3; :: thesis: verum

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: verum

end;

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)} or a in Subformulae (H '&' F) )
assume A2: a in ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)} ; :: thesis: a in Subformulae (H '&' F)
A3: ( not a in (Subformulae H) \/ (Subformulae F) or a in Subformulae H or a in Subformulae F ) by XBOOLE_0:def 3;

A4: now

assume a in Subformulae H ; :: thesis: a in Subformulae (H '&' F)
then consider G being Element of QC-WFF such that
A5: ( G = a & G is_subformula_of H ) by Def23;
H is_immediate_constituent_of H '&' F by Th62;
then H is_proper_subformula_of H '&' F by Th73;
then H is_subformula_of H '&' F by Def22;
then G is_subformula_of H '&' F by A5, Th77;
hence a in Subformulae (H '&' F) by A5, Def23; :: thesis: verum

end;

A6: now

assume a in Subformulae F ; :: thesis: a in Subformulae (H '&' F)
then consider G being Element of QC-WFF such that
A7: ( G = a & G is_subformula_of F ) by Def23;
F is_immediate_constituent_of H '&' F by Th62;
then F is_proper_subformula_of H '&' F by Th73;
then F is_subformula_of H '&' F by Def22;
then G is_subformula_of H '&' F by A7, Th77;
hence a in Subformulae (H '&' F) by A7, Def23; :: thesis: verum

end;

now

assume a in {(H '&' F)} ; :: thesis: a in Subformulae (H '&' F)
then ( a = H '&' F & H '&' F is_subformula_of H '&' F ) by TARSKI:def 1;
hence a in Subformulae (H '&' F) by Def23; :: thesis: verum

end;

hence a in Subformulae (H '&' F) by A2, A3, A4, A6, XBOOLE_0:def 3; :: thesis: verum