let FMT be non empty FMT_Space_Str ; :: thesis: for x being Element of FMT

for A being Subset of FMT holds

( x in A ^Fodelta iff for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) )

let x be Element of FMT; :: thesis: for A being Subset of FMT holds

( x in A ^Fodelta iff for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) )

let A be Subset of FMT; :: thesis: ( x in A ^Fodelta iff for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) )

thus ( x in A ^Fodelta implies for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) ) :: thesis: ( ( for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) ) implies x in A ^Fodelta )

( W meets A & W meets A ` ) ; :: thesis: x in A ^Fodelta

hence x in A ^Fodelta ; :: thesis: verum

for A being Subset of FMT holds

( x in A ^Fodelta iff for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) )

let x be Element of FMT; :: thesis: for A being Subset of FMT holds

( x in A ^Fodelta iff for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) )

let A be Subset of FMT; :: thesis: ( x in A ^Fodelta iff for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) )

thus ( x in A ^Fodelta implies for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) ) :: thesis: ( ( for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) ) implies x in A ^Fodelta )

proof

assume
for W being Subset of FMT st W in U_FMT x holds
assume
x in A ^Fodelta
; :: thesis: for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` )

then ex y being Element of FMT st

( y = x & ( for W being Subset of FMT st W in U_FMT y holds

( W meets A & W meets A ` ) ) ) ;

hence for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) ; :: thesis: verum

end;( W meets A & W meets A ` )

then ex y being Element of FMT st

( y = x & ( for W being Subset of FMT st W in U_FMT y holds

( W meets A & W meets A ` ) ) ) ;

hence for W being Subset of FMT st W in U_FMT x holds

( W meets A & W meets A ` ) ; :: thesis: verum

( W meets A & W meets A ` ) ; :: thesis: x in A ^Fodelta

hence x in A ^Fodelta ; :: thesis: verum