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<b>let </b><font color="Maroon">F</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="symsp_1.html#NM1">SymSp</a> of <font color="Maroon">F</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:51"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green">E31</font></i>: 
not <font color="Maroon">b</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 ;<br/>

<b>set </b><font color="Maroon">0F</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>;<br/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:51_1"/><b>then </b>
(  <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp;  <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> &amp;  <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> )
 <b>by </b><i>, , <a class="txt" href="symsp_1.html#E7"><i><font color="Green">Lemma40</font></i></a></i>;<br/><b>hence </b><a NAME="E3:51_1"/>
 <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 ;<br/><a NAME="E2:51_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> )
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:51_2"/><i><font color="Green">E34</font></i>: 
( not <font color="Maroon">p</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font> &amp; not <font color="Maroon">p</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/><a NAME="E5:51_2"/><i><font color="Green">E35</font></i>: 
 <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E6:51_2"/><i><font color="Green">E36</font></i>: 
 <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E7:51_2"/><i><font color="Green">E37</font></i>: 
 <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/><a NAME="E8:51_2"/>
 <a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:51_2"/>
 <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:51"/>
 <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/>


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