let A be complex-membered set ; :: thesis: for a being Complex holds a ** A = { (a * c) where c is Complex : c in A }
let a be Complex; :: thesis: a ** A = { (a * c) where c is Complex : c in A }
thus a ** A c= { (a * c) where c is Complex : c in A } :: according to XBOOLE_0:def 10 :: thesis: { (a * c) where c is Complex : c in A } c= a ** A
proof
let e be object ; :: according to TARSKI:def 3 :: thesis: ( not e in a ** A or e in { (a * c) where c is Complex : c in A } )
assume e in a ** A ; :: thesis: e in { (a * c) where c is Complex : c in A }
then consider c1, c2 being Complex such that
A1: e = c1 * c2 and
A2: c1 in {a} and
A3: c2 in A ;
c1 = a by A2, TARSKI:def 1;
hence e in { (a * c) where c is Complex : c in A } by A1, A3; :: thesis: verum
end;
let e be object ; :: according to TARSKI:def 3 :: thesis: ( not e in { (a * c) where c is Complex : c in A } or e in a ** A )
assume e in { (a * c) where c is Complex : c in A } ; :: thesis: e in a ** A
then ex c being Complex st
( e = a * c & c in A ) ;
hence e in a ** A by Th193; :: thesis: verum