let A, B be complex-membered set ; :: thesis: ( A = { (c " ) where c is Element of COMPLEX : c in B } implies B = { (c " ) where c is Element of COMPLEX : c in A } )
assume A1: A = { (c " ) where c is Element of COMPLEX : c in B } ; :: thesis: B = { (c " ) where c is Element of COMPLEX : c in A }
thus B c= { (c " ) where c is Element of COMPLEX : c in A } :: according to XBOOLE_0:def 10 :: thesis: { (c " ) where c is Element of COMPLEX : c in A } c= B

proof

let z be complex number ; :: according to MEMBERED:def 7 :: thesis: ( not z in B or z in { (c " ) where c is Element of COMPLEX : c in A } )
A2: z in COMPLEX by XCMPLX_0:def 2;
A3: ( z " in COMPLEX & z = (z " ) " ) by XCMPLX_0:def 2;
assume z in B ; :: thesis: z in { (c " ) where c is Element of COMPLEX : c in A }
then z " in A by A1, A2;
hence z in { (c " ) where c is Element of COMPLEX : c in A } by A3; :: thesis: verum

end;

let e be set ; :: according to TARSKI:def 3 :: thesis: ( not e in { (c " ) where c is Element of COMPLEX : c in A } or e in B )
assume e in { (c " ) where c is Element of COMPLEX : c in A } ; :: thesis: e in B
then consider r0 being Element of COMPLEX such that
A4: r0 " = e and
A5: r0 in A ;
ex c being Element of COMPLEX st
( c " = r0 & c in B ) by A1, A5;
hence e in B by A4; :: thesis: verum