let V be RealLinearSpace; :: thesis: for v1, v2 being VECTOR of V st {v1,v2} is linearly-independent holds

( v1 <> 0. V & v2 <> 0. V )

let v1, v2 be VECTOR of V; :: thesis: ( {v1,v2} is linearly-independent implies ( v1 <> 0. V & v2 <> 0. V ) )

A1: ( v1 in {v1,v2} & v2 in {v1,v2} ) by TARSKI:def 2;

assume {v1,v2} is linearly-independent ; :: thesis: ( v1 <> 0. V & v2 <> 0. V )

hence ( v1 <> 0. V & v2 <> 0. V ) by A1, Th6; :: thesis: verum

( v1 <> 0. V & v2 <> 0. V )

let v1, v2 be VECTOR of V; :: thesis: ( {v1,v2} is linearly-independent implies ( v1 <> 0. V & v2 <> 0. V ) )

A1: ( v1 in {v1,v2} & v2 in {v1,v2} ) by TARSKI:def 2;

assume {v1,v2} is linearly-independent ; :: thesis: ( v1 <> 0. V & v2 <> 0. V )

hence ( v1 <> 0. V & v2 <> 0. V ) by A1, Th6; :: thesis: verum