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 ^Fos iff ( x in A & ex V being Subset of FMT st
( V in U_FMT x & V \ {x} misses A ) ) )
let x be Element of FMT; :: thesis: for A being Subset of FMT holds
( x in A ^Fos iff ( x in A & ex V being Subset of FMT st
( V in U_FMT x & V \ {x} misses A ) ) )
let A be Subset of FMT; :: thesis: ( x in A ^Fos iff ( x in A & ex V being Subset of FMT st
( V in U_FMT x & V \ {x} misses A ) ) )
thus ( x in A ^Fos implies ( x in A & ex V being Subset of FMT st
( V in U_FMT x & V \ {x} misses A ) ) ) :: thesis: ( x in A & ex V being Subset of FMT st
( V in U_FMT x & V \ {x} misses A ) implies x in A ^Fos )

proof

assume x in A ^Fos ; :: thesis: ( x in A & ex V being Subset of FMT st
( V in U_FMT x & V \ {x} misses A ) )
then ex y being Element of FMT st
( y = x & y in A & ex V being Subset of FMT st
( V in U_FMT y & V \ {y} misses A ) ) ;
hence ( x in A & ex V being Subset of FMT st
( V in U_FMT x & V \ {x} misses A ) ) ; :: thesis: verum

end;

assume ( x in A & ex V being Subset of FMT st
( V in U_FMT x & V \ {x} misses A ) ) ; :: thesis: x in A ^Fos
hence x in A ^Fos ; :: thesis: verum