<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">K</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">K</font>;<br/>



<b>reconsider </b><font color="Maroon">W</font> =  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> #) as  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> =  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> as    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="vectsp_4.html#T50">VECTSP_4:50</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>;<br/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1"/>
not <font color="Maroon">V</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a></i>;<br/><a NAME="E4:9_1"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:9_1"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3">VECTSP_4:def 3</a></i>;<br/><b>hence </b><a NAME="E6:9_1"/>
<font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_2"/>
<font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Z</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:9_2"/><i><font color="Green">E52</font></i>: 
not <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T40">VECTSP_4:40</a></i>;<br/><a NAME="E4:9_2"/>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E5:9_2"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3">VECTSP_4:def 3</a></i>;<br/><a NAME="E6:9_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:9_2"/>
not <font color="Maroon">V</font> is <a href="realset2.html#V3">trivial</a>
 <b>by </b><i><a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:9"/>
( <font color="Maroon">V</font> is <a href="realset2.html#V3">trivial</a> iff  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">V</font> )
 <b>by </b><i/>;<br/>


</div>
