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<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>set </b><font color="Maroon">W</font> =  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font>;<br/>
<a NAME="E1:5"/><i><font color="Green">E43</font></i>: 
 <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <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> #)
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D4">VECTSP_4:def 4</a></i>;<br/>
<a NAME="E2:5"/><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:5"/>
 <a href="relat_1.html#K1">dom</a> the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:5"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span> &amp; the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U2">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>
<b>hence </b><a NAME="E5:5"/>
<font color="Maroon">V</font> is    <a href="vectsp_4.html#M1">Subspace</a> of  <a href="vectsp_4.html#K2">(Omega).</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>


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