consider I being finite Subset of V such that
A1: I is Basis of V by ZMODUL03:def 3;
assume rank V = 1 ; :: thesis: ex v being VECTOR of V st
( v <> 0. V & (Omega). V = Lin {v} )
then card I = 1 by A1, ZMODUL03:def 5;
then consider v being object such that
A2: I = {v} by CARD_2:42;
v in I by A2, TARSKI:def 1;
then reconsider v = v as VECTOR of V ;
A3: v <> 0. V by A1, A2, VECTSP_7:def 3;
Lin {v} = ModuleStr(# the carrier of V, the U5 of V, the ZeroF of V, the lmult of V #) by A1, A2, VECTSP_7:def 3;
hence ex v being VECTOR of V st
( v <> 0. V & (Omega). V = Lin {v} ) by A3; :: thesis: verum