let A be QC-alphabet ; :: thesis: for F, G, H being Element of QC-WFF A holds
( F is_immediate_constituent_of G '&' H iff ( F = G or F = H ) )
let F, G, H be Element of QC-WFF A; :: thesis: ( F is_immediate_constituent_of G '&' H iff ( F = G or F = H ) )
thus ( not F is_immediate_constituent_of G '&' H or F = G or F = H ) :: thesis: ( ( F = G or F = H ) implies F is_immediate_constituent_of G '&' H )

proof

G '&' H is conjunctive by QC_LANG1:def 20;
then A1: ((@ (G '&' H)) . 1) `1 = 2 by QC_LANG1:18;

A2: now :: thesis: not G '&' H = 'not' F

A3: 'not' F is negative by QC_LANG1:def 19;
assume G '&' H = 'not' F ; :: thesis: contradiction
hence contradiction by A1, A3, QC_LANG1:18; :: thesis: verum

end;

A4: now :: thesis: for x being bound_QC-variable of A holds not G '&' H = All (x,F)

given x being bound_QC-variable of A such that A5: G '&' H = All (x,F) ; :: thesis: contradiction
All (x,F) is universal by QC_LANG1:def 21;
hence contradiction by A1, A5, QC_LANG1:18; :: thesis: verum

end;

assume ( G '&' H = 'not' F or ex H1 being Element of QC-WFF A st
( G '&' H = F '&' H1 or G '&' H = H1 '&' F ) or ex x being bound_QC-variable of A st G '&' H = All (x,F) ) ; :: according to QC_LANG2:def 19 :: thesis: ( F = G or F = H )
hence ( F = G or F = H ) by A2, A4, Th2; :: thesis: verum

end;

assume ( F = G or F = H ) ; :: thesis: F is_immediate_constituent_of G '&' H
hence F is_immediate_constituent_of G '&' H by Def19; :: thesis: verum