let R be Ring; :: thesis: for V being LeftMod of R
for v1, v2 being Vector of V st not R is degenerated & {v1,v2} is linearly-independent holds
( v1 <> 0. V & v2 <> 0. V )
let V be LeftMod of R; :: thesis: for v1, v2 being Vector of V st not R is degenerated & {v1,v2} is linearly-independent holds
( v1 <> 0. V & v2 <> 0. V )
let v1, v2 be Vector of V; :: thesis: ( not R is degenerated & {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 ( not R is degenerated & {v1,v2} is linearly-independent ) ; :: thesis: ( v1 <> 0. V & v2 <> 0. V )
hence ( v1 <> 0. V & v2 <> 0. V ) by A1, Th2; :: thesis: verum