let A be complex-membered set ; :: thesis: for a being Complex holds A -- a = { (c - a) where c is Complex : c in A }
let a be Complex; :: thesis: A -- a = { (c - a) where c is Complex : c in A }
A1: A -- a = { (c1 - c2) where c1, c2 is Complex : ( c1 in A & c2 in {a} ) } by Th65;
thus A -- a c= { (c - a) where c is Complex : c in A } :: according to XBOOLE_0:def 10 :: thesis: { (c - a) 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 { (c - a) where c is Complex : c in A } )
assume e in A -- a ; :: thesis: e in { (c - a) where c is Complex : c in A }
then consider c1, c2 being Complex such that
A2: ( e = c1 - c2 & c1 in A ) and
A3: c2 in {a} by A1;
c2 = a by A3, TARSKI:def 1;
hence e in { (c - a) where c is Complex : c in A } by A2; :: thesis: verum

end;

let e be object ; :: according to TARSKI:def 3 :: thesis: ( not e in { (c - a) where c is Complex : c in A } or e in A -- a )
assume e in { (c - a) where c is Complex : c in A } ; :: thesis: e in A -- a
then ex c being Complex st
( e = c - a & c in A ) ;
hence e in A -- a by Th176; :: thesis: verum