let R be Ring; 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; 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; ( 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 )
; ( v1 <> 0. V & v2 <> 0. V )
hence
( v1 <> 0. V & v2 <> 0. V )
by A1, Th2; verum