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 ^Foi iff ex V being Subset of FMT st

( V in U_FMT x & V c= A ) )

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

( x in A ^Foi iff ex V being Subset of FMT st

( V in U_FMT x & V c= A ) )

let A be Subset of FMT; :: thesis: ( x in A ^Foi iff ex V being Subset of FMT st

( V in U_FMT x & V c= A ) )

thus ( x in A ^Foi implies ex V being Subset of FMT st

( V in U_FMT x & V c= A ) ) :: thesis: ( ex V being Subset of FMT st

( V in U_FMT x & V c= A ) implies x in A ^Foi )

( V in U_FMT x & V c= A ) ; :: thesis: x in A ^Foi

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

for A being Subset of FMT holds

( x in A ^Foi iff ex V being Subset of FMT st

( V in U_FMT x & V c= A ) )

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

( x in A ^Foi iff ex V being Subset of FMT st

( V in U_FMT x & V c= A ) )

let A be Subset of FMT; :: thesis: ( x in A ^Foi iff ex V being Subset of FMT st

( V in U_FMT x & V c= A ) )

thus ( x in A ^Foi implies ex V being Subset of FMT st

( V in U_FMT x & V c= A ) ) :: thesis: ( ex V being Subset of FMT st

( V in U_FMT x & V c= A ) implies x in A ^Foi )

proof

assume
ex V being Subset of FMT st
assume
x in A ^Foi
; :: thesis: ex V being Subset of FMT st

( V in U_FMT x & V c= A )

then ex y being Element of FMT st

( y = x & ex V being Subset of FMT st

( V in U_FMT y & V c= A ) ) ;

hence ex V being Subset of FMT st

( V in U_FMT x & V c= A ) ; :: thesis: verum

end;( V in U_FMT x & V c= A )

then ex y being Element of FMT st

( y = x & ex V being Subset of FMT st

( V in U_FMT y & V c= A ) ) ;

hence ex V being Subset of FMT st

( V in U_FMT x & V c= A ) ; :: thesis: verum

( V in U_FMT x & V c= A ) ; :: thesis: x in A ^Foi

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