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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">M</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="rusub_4.html#V2">Affine</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>



<b>consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:22"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<a NAME="E4:22"/>
( not <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> is <a href="rusub_4.html#V2">Affine</a> )
 
<b>by </b><i><a class="ref" href="rusub_4.html#T23">RUSUB_4:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">N</font> = <font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="rusub_4.html#V2">Affine</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<a NAME="E6:22"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">M</font> <a href="rusub_5.html#R1" target="_self">is_parallel_to</a> <font color="Maroon">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">v</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rusub_5.html#D1" target="_self">RUSUB_5:def 1</a><br/>

<b>thus </b><a NAME="E1:22_1"/>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="rusub_4.html#K6">+</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">L</font> = <font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <font color="Maroon">M</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="rusub_4.html#V2">Affine</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<a NAME="E8:22"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">L</font> is <a href="rusub_4.html#V3">Subspace-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<a NAME="E1:22_2"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:22_2"/>
<font color="Maroon">L</font> is <a href="rusub_4.html#V3">Subspace-like</a>
 <b>by </b><i><a class="ref" href="rusub_4.html#T26">RUSUB_4:26</a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon">N</font> = 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">L</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E10:22"/>
( <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="rusub_5.html#K1" target="_self">-</a> <font color="Maroon">M</font> &amp; <font color="Maroon">L</font> is <a href="rusub_4.html#V3">Subspace-like</a> &amp; <font color="Maroon">M</font> <a href="rusub_5.html#R1" target="_self">is_parallel_to</a> <font color="Maroon">L</font> )
 <b>by </b><i>, </i>;<br/>


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