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 ( P_A (x,A) = TRUE & ( for y being Element of FT holds
( not P_1 (x,y,A) = TRUE or not P_e (x,y) = FALSE ) ) ) )
let x be Element of FT; :: thesis: for A being Subset of FT holds
( x in A ^s iff ( P_A (x,A) = TRUE & ( for y being Element of FT holds
( not P_1 (x,y,A) = TRUE or not P_e (x,y) = FALSE ) ) ) )
let A be Subset of FT; :: thesis: ( x in A ^s iff ( P_A (x,A) = TRUE & ( for y being Element of FT holds
( not P_1 (x,y,A) = TRUE or not P_e (x,y) = FALSE ) ) ) )
A1: ( x in A ^s implies ( P_A (x,A) = TRUE & ( for y being Element of FT holds
( not P_1 (x,y,A) = TRUE or not P_e (x,y) = FALSE ) ) ) )

proof

assume A2: x in A ^s ; :: thesis: ( P_A (x,A) = TRUE & ( for y being Element of FT holds
( not P_1 (x,y,A) = TRUE or not P_e (x,y) = FALSE ) ) )
then (U_FT x) \ {x} misses A by FIN_TOPO:9;
then A3: ((U_FT x) \ {x}) /\ A = {} ;
A4: for y being Element of FT holds
( not P_1 (x,y,A) = TRUE or not P_e (x,y) = FALSE )

proof

given y being Element of FT such that A5: P_1 (x,y,A) = TRUE and
A6: P_e (x,y) = FALSE ; :: thesis: contradiction
not x = y by A6, Def5;
then A7: not y in {x} by TARSKI:def 1;
y in U_FT x by A5, Def1;
then A8: y in (U_FT x) \ {x} by A7, XBOOLE_0:def 5;
y in A by A5, Def1;
hence contradiction by A3, A8, XBOOLE_0:def 4; :: thesis: verum

end;

x in A by A2, FIN_TOPO:9;
hence ( P_A (x,A) = TRUE & ( for y being Element of FT holds
( not P_1 (x,y,A) = TRUE or not P_e (x,y) = FALSE ) ) ) by A4, Def4; :: thesis: verum

end;

( P_A (x,A) = TRUE & ( for y being Element of FT holds
( not P_1 (x,y,A) = TRUE or not P_e (x,y) = FALSE ) ) implies x in A ^s )

proof

assume that
A9: P_A (x,A) = TRUE and
A10: for y being Element of FT holds
( not P_1 (x,y,A) = TRUE or not P_e (x,y) = FALSE ) ; :: thesis: x in A ^s
for y being Element of FT holds not y in ((U_FT x) \ {x}) /\ A

proof

let y be Element of FT; :: thesis: not y in ((U_FT x) \ {x}) /\ A
( not P_1 (x,y,A) = TRUE or not P_e (x,y) = FALSE ) by A10;
then ( not y in U_FT x or x = y or not y in A ) by Def1, Def5;
then ( not y in U_FT x or y in {x} or not y in A ) by TARSKI:def 1;
then ( not y in (U_FT x) \ {x} or not y in A ) by XBOOLE_0:def 5;
hence not y in ((U_FT x) \ {x}) /\ A by XBOOLE_0:def 4; :: thesis: verum

end;

then ((U_FT x) \ {x}) /\ A = {} by SUBSET_1:4;
then A11: (U_FT x) \ {x} misses A ;
x in A by A9, Def4;
hence x in A ^s by A11; :: thesis: verum

end;

hence ( x in A ^s iff ( P_A (x,A) = TRUE & ( for y being Element of FT holds
( not P_1 (x,y,A) = TRUE or not P_e (x,y) = FALSE ) ) ) ) by A1; :: thesis: verum