let p be Element of CQC-WFF ; :: thesis: for X being Subset of CQC-WFF holds
( X |- 'not' ('not' p) iff X |- p )
let X be Subset of CQC-WFF; :: thesis: ( X |- 'not' ('not' p) iff X |- p )
thus ( X |- 'not' ('not' p) implies X |- p ) :: thesis: ( X |- p implies X |- 'not' ('not' p) )

proof

assume A1: X |- 'not' ('not' p) ; :: thesis: X |- p
X |- ('not' ('not' p)) => p by CQC_THE1:59;
hence X |- p by A1, CQC_THE1:55; :: thesis: verum

end;

assume A2: X |- p ; :: thesis: X |- 'not' ('not' p)
X |- p => ('not' ('not' p)) by CQC_THE1:59;
hence X |- 'not' ('not' p) by A2, CQC_THE1:55; :: thesis: verum