let V be RealLinearSpace; :: thesis: for v being VECTOR of V holds
( {v} is linearly-independent iff v <> 0. V )
let v be VECTOR of V; :: thesis: ( {v} is linearly-independent iff v <> 0. V )
thus ( {v} is linearly-independent implies v <> 0. V ) :: thesis: ( v <> 0. V implies {v} is linearly-independent )

proof

assume {v} is linearly-independent ; :: thesis: v <> 0. V
then not 0. V in {v} by Th6;
hence v <> 0. V by TARSKI:def 1; :: thesis: verum

end;

assume A1: v <> 0. V ; :: thesis: {v} is linearly-independent
let l be Linear_Combination of {v}; :: according to RLVECT_3:def 1 :: thesis: ( Sum l = 0. V implies Carrier l = {} )
A2: Carrier l c= {v} by RLVECT_2:def 6;
assume A3: Sum l = 0. V ; :: thesis: Carrier l = {}

now :: thesis: Carrier l = {}

per cases ( Carrier l = {} or Carrier l = {v} ) by A2, ZFMISC_1:33;

suppose Carrier l = {} ; :: thesis: Carrier l = {}

hence Carrier l = {} ; :: thesis: verum

end;

suppose A4: Carrier l = {v} ; :: thesis: Carrier l = {}

A5: 0. V = (l . v) * v by A3, RLVECT_2:32;

now :: thesis: not v in Carrier l

assume v in Carrier l ; :: thesis: contradiction
then ex u being VECTOR of V st
( v = u & l . u <> 0 ) ;
hence contradiction by A1, A5, RLVECT_1:11; :: thesis: verum

end;

hence Carrier l = {} by A4, TARSKI:def 1; :: thesis: verum

end;

end;

end;

hence Carrier l = {} ; :: thesis: verum