let FMT be non empty FMT_Space_Str ; :: thesis:
let A be Subset of FMT; :: thesis:
for x being object holds
( x in ((A `) ^Fob) ` iff x in A ^Foi )

let x be object ; :: thesis:
thus ( x in ((A `) ^Fob) ` implies x in A ^Foi ) :: thesis:

assume A1: x in ((A `) ^Fob) ` ; :: thesis:
then reconsider y = x as Element of FMT ;
not y in (A `) ^Fob by A1, XBOOLE_0:def 5;
then consider V being Subset of FMT such that
A2: V in U_FMT y and
A3: V misses A ` ;
V /\ (A `) = {} by A3;
then V \ A = {} by SUBSET_1:13;
then V c= A by XBOOLE_1:37;
hence x in A ^Foi by A2; :: thesis:

end;

assume A4: x in A ^Foi ; :: thesis:
then reconsider y = x as Element of FMT ;
consider V being Subset of FMT such that
A5: V in U_FMT y and
A6: V c= A by A4, Th21;
V \ A = {} by A6, XBOOLE_1:37;
then V /\ (A `) = {} by SUBSET_1:13;
then V misses A ` ;
then not y in (A `) ^Fob by A5, Th20;
hence x in ((A `) ^Fob) ` by XBOOLE_0:def 5; :: thesis:

end;