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 )
proof
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;
assume ex V being Subset of FMT st
( V in U_FMT x & V c= A ) ; :: thesis: x in A ^Foi
hence x in A ^Foi ; :: thesis: verum