let V be finite-dimensional RealUnitarySpace; :: thesis: ( dim V = 1 iff ex v being VECTOR of V st
( v <> 0. V & (Omega). V = Lin {v} ) )

hereby :: thesis: ( ex v being VECTOR of V st
( v <> 0. V & (Omega). V = Lin {v} ) implies dim V = 1 )
consider I being finite Subset of V such that
A1: I is Basis of V by Def1;
assume dim V = 1 ; :: thesis: ex v being VECTOR of V st
( v <> 0. V & (Omega). V = Lin {v} )

then card I = 1 by ;
then consider v being object such that
A2: I = {v} by CARD_2:42;
v in I by ;
then reconsider v = v as VECTOR of V ;
{v} is linearly-independent by ;
then A3: v <> 0. V by RLVECT_3:8;
Lin {v} = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by ;
hence ex v being VECTOR of V st
( v <> 0. V & (Omega). V = Lin {v} ) by ; :: thesis: verum
end;
given v being VECTOR of V such that A4: ( v <> 0. V & (Omega). V = Lin {v} ) ; :: thesis: dim V = 1
( {v} is linearly-independent & Lin {v} = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) ) by ;
then A5: {v} is Basis of V by RUSUB_3:def 2;
card {v} = 1 by CARD_1:30;
hence dim V = 1 by ; :: thesis: verum