let FT be non empty RelStr ; :: thesis: for x being Element of FT

for A being Subset of FT holds

( x in A ^b iff U_FT x meets A )

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

( x in A ^b iff U_FT x meets A )

let A be Subset of FT; :: thesis: ( x in A ^b iff U_FT x meets A )

thus ( x in A ^b implies U_FT x meets A ) :: thesis: ( U_FT x meets A implies x in A ^b )

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

for A being Subset of FT holds

( x in A ^b iff U_FT x meets A )

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

( x in A ^b iff U_FT x meets A )

let A be Subset of FT; :: thesis: ( x in A ^b iff U_FT x meets A )

thus ( x in A ^b implies U_FT x meets A ) :: thesis: ( U_FT x meets A implies x in A ^b )

proof

assume
U_FT x meets A
; :: thesis: x in A ^b
assume
x in A ^b
; :: thesis: U_FT x meets A

then ex y being Element of FT st

( y = x & U_FT y meets A ) ;

hence U_FT x meets A ; :: thesis: verum

end;then ex y being Element of FT st

( y = x & U_FT y meets A ) ;

hence U_FT x meets A ; :: thesis: verum

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