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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  ex <font color="Olive" title="b2">L</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b2">L</font> is <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a> &amp; <font color="Olive" title="b1">M</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Olive" title="b2">L</font> )</span><br/><b>let </b><font color="Maroon" title="c2">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">L</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b1">L</font> is <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a> &amp; <font color="Maroon" title="c2">M</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Olive" title="b1">L</font> )</span><br/>



<b>consider </b><font color="Maroon" title="c3">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:22"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">v</font> = <font color="Maroon" title="c3">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>
<a NAME="E4:22"/>
( not <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> )
 
<b>by </b><i><a class="ref" href="rusub_4.html#T23" title="RUSUB_4:th.23">RUSUB_4:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">N</font> = <font color="Maroon" title="c2">M</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_5.html#T4" target="_self" title="RUSUB_5:th.4">Th4</a>, <a class="ref" href="rusub_5.html#T8" target="_self" title="RUSUB_5:th.8">Th8</a></i>;<br/>
<a NAME="E6:22"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Maroon" title="c5">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" title="c4">v</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rusub_5.html#D1" target="_self" title="RUSUB_5:def.1">RUSUB_5:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">N</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<b>thus </b><a NAME="E1:22_1"/>
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">N</font> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="rusub_5.html#T9" target="_self" title="RUSUB_5:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c6">L</font> = <font color="Maroon" title="c2">M</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c2">M</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="rusub_5.html#T4" target="_self" title="RUSUB_5:th.4">Th4</a>, <a class="ref" href="rusub_5.html#T8" target="_self" title="RUSUB_5:th.8">Th8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">L</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">L</font> is <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a> &amp; <font color="Maroon" title="c2">M</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Maroon" title="c6">L</font> )</span><br/>

<a NAME="E8:22"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c6">L</font> is <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">L</font>
 
<b>by </b><i><a class="ref" href="rusub_5.html#T12" target="_self" title="RUSUB_5:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E2:22_2"/>
<font color="Maroon" title="c6">L</font> is <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a>
 <b>by </b><i><a class="ref" href="rusub_4.html#T26" title="RUSUB_4:th.26">RUSUB_4:26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<font color="Maroon" title="c5">N</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> </span>}</span> 
<b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="rusub_5.html#T19" target="_self" title="RUSUB_5:th.19">Th19</a></i>
<br/>.= 
<font color="Maroon" title="c6">L</font>
<b>by </b><i><a class="ref" href="rusub_5.html#T18" target="_self" title="RUSUB_5:th.18">Th18</a></i>
;<br/>
<b>hence </b><a NAME="E10:22"/>
( <font color="Maroon" title="c6">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> <a href="rusub_5.html#K1" title="RUSUB_5:func.1">-</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">L</font> is <a href="rusub_4.html#V3" title="RUSUB_4:attr.3">Subspace-like</a> &amp; <font color="Maroon" title="c2">M</font> <a href="rusub_5.html#R1" title="RUSUB_5:pred.1">is_parallel_to</a> <font color="Maroon" title="c6">L</font> )
 <b>by </b><i><a class="txt" href="#E6:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E8:22"><i><font color="Green" title="E17">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
