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

proof

assume x in A ^Fon ; :: thesis: ( x in A & ( for V being Subset of FMT st V in U_FMT x holds
V \ {x} meets A ) )
then ( x in A & not x in A ^Fos ) by XBOOLE_0:def 5;
hence ( x in A & ( for V being Subset of FMT st V in U_FMT x holds
V \ {x} meets A ) ) ; :: thesis: verum

end;

assume that
A1: x in A and
A2: for V being Subset of FMT st V in U_FMT x holds
V \ {x} meets A ; :: thesis: x in A ^Fon
not x in A ^Fos by A2, Th22;
hence x in A ^Fon by A1, XBOOLE_0:def 5; :: thesis: verum