let R be Ring; :: thesis: for V being RightMod of R
for v1, v2 being Vector of V st 0. R <> 1_ R & {v1,v2} is linearly-independent holds
( v1 <> 0. V & v2 <> 0. V )
let V be RightMod of R; :: thesis: for v1, v2 being Vector of V st 0. R <> 1_ R & {v1,v2} is linearly-independent holds
( v1 <> 0. V & v2 <> 0. V )
let v1, v2 be Vector of V; :: thesis: ( 0. R <> 1_ R & {v1,v2} is linearly-independent implies ( v1 <> 0. V & v2 <> 0. V ) )
assume that
A1: 0. R <> 1_ R and
A2: {v1,v2} is linearly-independent ; :: thesis: ( v1 <> 0. V & v2 <> 0. V )
( v1 in {v1,v2} & v2 in {v1,v2} ) by TARSKI:def 2;
hence ( v1 <> 0. V & v2 <> 0. V ) by A1, A2, Th3; :: thesis: verum