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<b>let </b><font color="Maroon">GF</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">w</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_7.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E2:18_1"/><b>then </b>
<font color="Maroon">w</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">L</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/><a NAME="E5:18_1"/>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D7">VECTSP_6:def 7</a></i>;<br/><a NAME="E6:18_1"/><b>then </b>
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then consider </b><font color="Maroon">K</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E8:18_1"/><i><font color="Green">E42</font></i>: 
(  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> &amp;  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">K</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E9:18_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E3:18"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:18"/>
 <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font> is    <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#T35">VECTSP_4:35</a></i>;<br/>


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