let FT be non empty RelStr ; :: thesis: for x being Element of FT
for A being Subset of FT holds
( x in A ^s iff ( x in A & (U_FT x) \ {x} misses A ) )
let x be Element of FT; :: thesis: for A being Subset of FT holds
( x in A ^s iff ( x in A & (U_FT x) \ {x} misses A ) )
let A be Subset of FT; :: thesis: ( x in A ^s iff ( x in A & (U_FT x) \ {x} misses A ) )
thus ( x in A ^s implies ( x in A & (U_FT x) \ {x} misses A ) ) :: thesis: ( x in A & (U_FT x) \ {x} misses A implies x in A ^s )

proof

assume x in A ^s ; :: thesis: ( x in A & (U_FT x) \ {x} misses A )
then ex y being Element of FT st
( y = x & y in A & (U_FT y) \ {y} misses A ) ;
hence ( x in A & (U_FT x) \ {x} misses A ) ; :: thesis: verum

end;

assume ( x in A & (U_FT x) \ {x} misses A ) ; :: thesis: x in A ^s
hence x in A ^s ; :: thesis: verum