let V be RealUnitarySpace; :: thesis: for W being Subspace of V
for u, v being VECTOR of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )
let W be Subspace of V; :: thesis: for u, v being VECTOR of V holds
( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )
let u, v be VECTOR of V; :: thesis: ( u in v + W iff ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) )
thus ( u in v + W implies ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) ) :: thesis: ( ex v1 being VECTOR of V st
( v1 in W & u = v - v1 ) implies u in v + W )

proof

assume u in v + W ; :: thesis: ex v1 being VECTOR of V st
( v1 in W & u = v - v1 )
then consider v1 being VECTOR of V such that
A1: u = v + v1 and
A2: v1 in W ;
take x = - v1; :: thesis: ( x in W & u = v - x )
thus x in W by A2, Th16; :: thesis: u = v - x
thus u = v - x by A1, RLVECT_1:17; :: thesis: verum

end;

given v1 being VECTOR of V such that A3: v1 in W and
A4: u = v - v1 ; :: thesis: u in v + W
- v1 in W by A3, Th16;
hence u in v + W by A4; :: thesis: verum