set Y = { (v + u) where u is VECTOR of V : u in W } ;
defpred S₁[ object ] means ex u being VECTOR of V st
( $1 = v + u & u in W );
consider X being set such that
A1: for x being object holds
( x in X iff ( x in the carrier of V & S₁[x] ) ) from XBOOLE_0:sch 1();
X c= the carrier of V by A1;
then reconsider X = X as Subset of V ;
A2: { (v + u) where u is VECTOR of V : u in W } c= X

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { (v + u) where u is VECTOR of V : u in W } or x in X )
assume x in { (v + u) where u is VECTOR of V : u in W } ; :: thesis: x in X
then ex u being VECTOR of V st
( x = v + u & u in W ) ;
hence x in X by A1; :: thesis: verum

end;

X c= { (v + u) where u is VECTOR of V : u in W }

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in { (v + u) where u is VECTOR of V : u in W } )
assume x in X ; :: thesis: x in { (v + u) where u is VECTOR of V : u in W }
then ex u being VECTOR of V st
( x = v + u & u in W ) by A1;
hence x in { (v + u) where u is VECTOR of V : u in W } ; :: thesis: verum

end;

hence { (v + u) where u is VECTOR of V : u in W } is Subset of V by A2, XBOOLE_0:def 10; :: thesis: verum