let FMT be non empty FMT_Space_Str ; :: thesis: for A being Subset of FMT holds A ^Fodelta = (A ^Fodel_i ) \/ (A ^Fodel_o )
let A be Subset of FMT; :: thesis: A ^Fodelta = (A ^Fodel_i ) \/ (A ^Fodel_o )
for x being set holds
( x in A ^Fodelta iff x in (A ^Fodel_i ) \/ (A ^Fodel_o ) )

proof

let x be set ; :: thesis: ( x in A ^Fodelta iff x in (A ^Fodel_i ) \/ (A ^Fodel_o ) )
thus ( x in A ^Fodelta implies x in (A ^Fodel_i ) \/ (A ^Fodel_o ) ) :: thesis: ( x in (A ^Fodel_i ) \/ (A ^Fodel_o ) implies x in A ^Fodelta )

proof

assume x in A ^Fodelta ; :: thesis: x in (A ^Fodel_i ) \/ (A ^Fodel_o )
then x in ([#] the carrier of FMT) /\ (A ^Fodelta ) by XBOOLE_1:28;
then x in (A \/ (A ` )) /\ (A ^Fodelta ) by SUBSET_1:25;
hence x in (A ^Fodel_i ) \/ (A ^Fodel_o ) by XBOOLE_1:23; :: thesis: verum

end;

assume x in (A ^Fodel_i ) \/ (A ^Fodel_o ) ; :: thesis: x in A ^Fodelta
then x in (A \/ (A ` )) /\ (A ^Fodelta ) by XBOOLE_1:23;
then x in ([#] the carrier of FMT) /\ (A ^Fodelta ) by SUBSET_1:25;
hence x in A ^Fodelta by XBOOLE_1:28; :: thesis: verum

end;

hence A ^Fodelta = (A ^Fodel_i ) \/ (A ^Fodel_o ) by TARSKI:2; :: thesis: verum