reserve x for set;
reserve a,b,c,d,e,r1,r2,r3,r4,r5,r6 for Real;
reserve V for RealLinearSpace;
reserve u,v,v1,v2,v3,w,w1,w2,w3 for VECTOR of V;
reserve W,W1,W2 for Subspace of V;

theorem
  v in Lin{w} & v <> 0.V implies Lin{v} = Lin{w}
proof
  assume that
A1: v in Lin{w} and
A2: v <> 0.V;
  consider a such that
A3: v = a * w by A1,Th8;
  now
    let u;
A4: a <> 0 by A2,A3,RLVECT_1:10;
    thus u in Lin{v} implies u in Lin{w}
    proof
      assume u in Lin{v};
      then consider b such that
A5:   u = b * v by Th8;
      u = b * a * w by A3,A5,RLVECT_1:def 7;
      hence thesis by Th8;
    end;
    assume u in Lin{w};
    then consider b such that
A6: u = b * w by Th8;
    a" * v = a" * a * w by A3,RLVECT_1:def 7
      .= 1 * w by A4,XCMPLX_0:def 7
      .= w by RLVECT_1:def 8;
    then u = b * a" * v by A6,RLVECT_1:def 7;
    hence u in Lin{v} by Th8;
  end;
  hence thesis by RLSUB_1:31;
end;
