let H, F be ZF-formula; :: thesis: Subformulae (H '&' F) = ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)}

now

let a be set ; :: thesis: ( ( a in Subformulae (H '&' F) implies a in ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)} ) & ( a in ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)} implies a in Subformulae (H '&' F) ) )
thus ( a in Subformulae (H '&' F) implies a in ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)} ) :: thesis: ( a in ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)} implies a in Subformulae (H '&' F) )

proof

assume a in Subformulae (H '&' F) ; :: thesis: a in ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)}
then consider G being ZF-formula such that
A1: ( G = a & G is_subformula_of H '&' F ) by Def42;

now

assume G <> H '&' F ; :: thesis: a in (Subformulae H) \/ (Subformulae F)
then G is_proper_subformula_of H '&' F by A1, Def41;
then ( G is_subformula_of H or G is_subformula_of F ) by Th91;
then ( a in Subformulae H or a in Subformulae F ) by A1, Def42;
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;

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 ZF-formula such that
A5: ( G = a & G is_subformula_of H ) by Def42;
H is_immediate_constituent_of H '&' F by Th72;
then H is_proper_subformula_of H '&' F by Th82;
then H is_subformula_of H '&' F by Def41;
then G is_subformula_of H '&' F by A5, Th86;
hence a in Subformulae (H '&' F) by A5, Def42; :: thesis: verum

end;

A6: now

assume a in Subformulae F ; :: thesis: a in Subformulae (H '&' F)
then consider G being ZF-formula such that
A7: ( G = a & G is_subformula_of F ) by Def42;
F is_immediate_constituent_of H '&' F by Th72;
then F is_proper_subformula_of H '&' F by Th82;
then F is_subformula_of H '&' F by Def41;
then G is_subformula_of H '&' F by A7, Th86;
hence a in Subformulae (H '&' F) by A7, Def42; :: 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 Th79, TARSKI:def 1;
hence a in Subformulae (H '&' F) by Def42; :: thesis: verum

end;

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

end;

hence Subformulae (H '&' F) = ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)} by TARSKI:2; :: thesis: verum