let A be QC-alphabet ; :: thesis: for x being bound_QC-variable of A
for p being QC-formula of A holds
( Ex (x,p) is closed iff still_not-bound_in p c= {x} )
let x be bound_QC-variable of A; :: thesis: for p being QC-formula of A holds
( Ex (x,p) is closed iff still_not-bound_in p c= {x} )
let p be QC-formula of A; :: thesis: ( Ex (x,p) is closed iff still_not-bound_in p c= {x} )
thus ( Ex (x,p) is closed implies still_not-bound_in p c= {x} ) :: thesis: ( still_not-bound_in p c= {x} implies Ex (x,p) is closed )

proof

assume still_not-bound_in (Ex (x,p)) = {} ; :: according to QC_LANG1:def 31 :: thesis: still_not-bound_in p c= {x}
then {} = (still_not-bound_in p) \ {x} by Th19;
hence still_not-bound_in p c= {x} by XBOOLE_1:37; :: thesis: verum

end;

assume still_not-bound_in p c= {x} ; :: thesis: Ex (x,p) is closed
then {} = (still_not-bound_in p) \ {x} by XBOOLE_1:37;
hence still_not-bound_in (Ex (x,p)) = {} by Th19; :: according to QC_LANG1:def 31 :: thesis: verum