let G be Group; :: thesis: for A being non empty Subset of G

for N being Subgroup of G holds N ~ (N ~ A) = N ` (N ~ A)

let A be non empty Subset of G; :: thesis: for N being Subgroup of G holds N ~ (N ~ A) = N ` (N ~ A)

let N be Subgroup of G; :: thesis: N ~ (N ~ A) = N ` (N ~ A)

thus N ~ (N ~ A) c= N ` (N ~ A) :: according to XBOOLE_0:def 10 :: thesis: N ` (N ~ A) c= N ~ (N ~ A)

assume A5: x in N ` (N ~ A) ; :: thesis: x in N ~ (N ~ A)

then reconsider x = x as Element of G ;

A6: x * N c= N ~ A by A5, Th31;

x in x * N by GROUP_2:108;

then x * N meets N ~ A by A6, XBOOLE_0:3;

hence x in N ~ (N ~ A) ; :: thesis: verum

for N being Subgroup of G holds N ~ (N ~ A) = N ` (N ~ A)

let A be non empty Subset of G; :: thesis: for N being Subgroup of G holds N ~ (N ~ A) = N ` (N ~ A)

let N be Subgroup of G; :: thesis: N ~ (N ~ A) = N ` (N ~ A)

thus N ~ (N ~ A) c= N ` (N ~ A) :: according to XBOOLE_0:def 10 :: thesis: N ` (N ~ A) c= N ~ (N ~ A)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in N ` (N ~ A) or x in N ~ (N ~ A) )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in N ~ (N ~ A) or x in N ` (N ~ A) )

assume A1: x in N ~ (N ~ A) ; :: thesis: x in N ` (N ~ A)

then reconsider x = x as Element of G ;

x * N meets N ~ A by A1, Th32;

then consider z being object such that

A2: ( z in x * N & z in N ~ A ) by XBOOLE_0:3;

reconsider z = z as Element of G by A2;

z * N meets A by A2, Th14;

then A3: x * N meets A by A2, Th2;

x * N c= N ~ A

end;assume A1: x in N ~ (N ~ A) ; :: thesis: x in N ` (N ~ A)

then reconsider x = x as Element of G ;

x * N meets N ~ A by A1, Th32;

then consider z being object such that

A2: ( z in x * N & z in N ~ A ) by XBOOLE_0:3;

reconsider z = z as Element of G by A2;

z * N meets A by A2, Th14;

then A3: x * N meets A by A2, Th2;

x * N c= N ~ A

proof

hence
x in N ` (N ~ A)
; :: thesis: verum
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in x * N or y in N ~ A )

assume A4: y in x * N ; :: thesis: y in N ~ A

then reconsider y = y as Element of G ;

x * N = y * N by A4, Th2;

hence y in N ~ A by A3; :: thesis: verum

end;assume A4: y in x * N ; :: thesis: y in N ~ A

then reconsider y = y as Element of G ;

x * N = y * N by A4, Th2;

hence y in N ~ A by A3; :: thesis: verum

assume A5: x in N ` (N ~ A) ; :: thesis: x in N ~ (N ~ A)

then reconsider x = x as Element of G ;

A6: x * N c= N ~ A by A5, Th31;

x in x * N by GROUP_2:108;

then x * N meets N ~ A by A6, XBOOLE_0:3;

hence x in N ~ (N ~ A) ; :: thesis: verum