let V be RealLinearSpace; :: thesis: for v being VECTOR of V
for A being Subset of V st v + A is affinely-independent holds
A is affinely-independent
let v be VECTOR of V; :: thesis: for A being Subset of V st v + A is affinely-independent holds
A is affinely-independent
let A be Subset of V; :: thesis: ( v + A is affinely-independent implies A is affinely-independent )
assume A1: v + A is affinely-independent ; :: thesis: A is affinely-independent
(- v) + (v + A) = ((- v) + v) + A by Th5
.= (0. V) + A by RLVECT_1:5
.= A by Th6 ;
hence A is affinely-independent by A1; :: thesis: verum