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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="rusub_4.html#V2" target="_self">Affine</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E1:43"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> is <a href="rusub_4.html#V3" target="_self">Subspace-like</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:43"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">x</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="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:43_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="rusub_3.html#T2">RUSUB_3:2</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1"/>
<font color="Maroon">x</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="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:43"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</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="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:43"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</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="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:43_2"/>
<font color="Maroon">x</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="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E2:43_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="rusub_3.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="rlvect_2.html#M2">Linear_Combination</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:43_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T1">RUSUB_3:1</a></i>;<br/>
<a NAME="E5:43_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:43_2"/>
for <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:43_2"/><b>then </b>
<font color="Maroon">A</font> is <a href="rlsub_1.html#V1">lineary-closed</a>
 
<b>by </b><i><a class="txt" href="rusub_4.html#E4:43"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rlsub_1.html#D1">RLSUB_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:43_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="rusub_4.html#E2:43"><i><font color="Green">E26</font></i></a>, <a class="ref" href="rlvect_2.html#T47">RLVECT_2:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:43"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:43"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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