let G be commutative Group; :: thesis: for A, B being Subset of G holds (A * B) " = (A ") * (B ")
let A, B be Subset of G; :: thesis: (A * B) " = (A ") * (B ")
thus (A * B) " c= (A ") * (B ") :: according to XBOOLE_0:def 10 :: thesis: (A ") * (B ") c= (A * B) "
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (A * B) " or x in (A ") * (B ") )
assume x in (A * B) " ; :: thesis: x in (A ") * (B ")
then consider g being Element of G such that
A1: x = g " and
A2: g in A * B ;
consider g1, g2 being Element of G such that
A3: g = g1 * g2 and
A4: ( g1 in A & g2 in B ) by A2;
A5: ( g1 " in A " & g2 " in B " ) by A4;
x = (g1 ") * (g2 ") by ;
hence x in (A ") * (B ") by A5; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (A ") * (B ") or x in (A * B) " )
assume x in (A ") * (B ") ; :: thesis: x in (A * B) "
then consider g1, g2 being Element of G such that
A6: x = g1 * g2 and
A7: g1 in A " and
A8: g2 in B " ;
consider b being Element of G such that
A9: g2 = b " and
A10: b in B by A8;
consider a being Element of G such that
A11: g1 = a " and
A12: a in A by A7;
A13: a * b in A * B by ;
x = (a * b) " by ;
hence x in (A * B) " by A13; :: thesis: verum