let FT be non empty RelStr ; :: thesis: for A being Subset of FT holds A ^delta = (A ^b) /\ ((A `) ^b)

let A be Subset of FT; :: thesis: A ^delta = (A ^b) /\ ((A `) ^b)

for x being object holds

( x in A ^delta iff x in (A ^b) /\ ((A `) ^b) )

let A be Subset of FT; :: thesis: A ^delta = (A ^b) /\ ((A `) ^b)

for x being object holds

( x in A ^delta iff x in (A ^b) /\ ((A `) ^b) )

proof

hence
A ^delta = (A ^b) /\ ((A `) ^b)
by TARSKI:2; :: thesis: verum
let x be object ; :: thesis: ( x in A ^delta iff x in (A ^b) /\ ((A `) ^b) )

thus ( x in A ^delta implies x in (A ^b) /\ ((A `) ^b) ) :: thesis: ( x in (A ^b) /\ ((A `) ^b) implies x in A ^delta )

then reconsider y = x as Element of FT ;

x in (A `) ^b by A3, XBOOLE_0:def 4;

then A4: U_FT y meets A ` by Th8;

x in A ^b by A3, XBOOLE_0:def 4;

then U_FT y meets A by Th8;

hence x in A ^delta by A4; :: thesis: verum

end;thus ( x in A ^delta implies x in (A ^b) /\ ((A `) ^b) ) :: thesis: ( x in (A ^b) /\ ((A `) ^b) implies x in A ^delta )

proof

assume A3:
x in (A ^b) /\ ((A `) ^b)
; :: thesis: x in A ^delta
assume A1:
x in A ^delta
; :: thesis: x in (A ^b) /\ ((A `) ^b)

then reconsider y = x as Element of FT ;

U_FT y meets A ` by A1, Th5;

then A2: x in (A `) ^b ;

U_FT y meets A by A1, Th5;

then x in A ^b ;

hence x in (A ^b) /\ ((A `) ^b) by A2, XBOOLE_0:def 4; :: thesis: verum

end;then reconsider y = x as Element of FT ;

U_FT y meets A ` by A1, Th5;

then A2: x in (A `) ^b ;

U_FT y meets A by A1, Th5;

then x in A ^b ;

hence x in (A ^b) /\ ((A `) ^b) by A2, XBOOLE_0:def 4; :: thesis: verum

then reconsider y = x as Element of FT ;

x in (A `) ^b by A3, XBOOLE_0:def 4;

then A4: U_FT y meets A ` by Th8;

x in A ^b by A3, XBOOLE_0:def 4;

then U_FT y meets A by Th8;

hence x in A ^delta by A4; :: thesis: verum