<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="symsp_1.html#NM1">SymSp</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>











<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E44</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E3:48"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="symsp_1.html#E2:48"><i><font color="Green">E44</font></i></a>, <a class="txt" href="symsp_1.html#E62"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green">E46</font></i>: 
(  <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>9</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>8</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E47</font></i>: 
 <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E2:48_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E3:48"><i><font color="Green">E45</font></i></a>, <a class="ref" href="symsp_1.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E62"><i><font color="Green">Lemma41</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><a NAME="E1:48_1_2"/><i><font color="Green">E49</font></i>: 
( not <font color="Maroon">c<sub>11</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>7</sub></font> &amp; not <font color="Maroon">c<sub>11</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E2:48_1_2"/>
<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">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48"><i><font color="Green">E43</font></i></a>, <a class="txt" href="symsp_1.html#E2:48"><i><font color="Green">E44</font></i></a>, <a class="txt" href="symsp_1.html#E1:48_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="symsp_1.html#E1:48_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="symsp_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E3:48_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>;<br/><a NAME="E4:48_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_2_1"/><i><font color="Green">E51</font></i>: 
 <a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:48_1_2_1"/>
not <font color="Maroon">c<sub>7</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="symsp_1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E3:48_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48_1_2_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="symsp_1.html#T36" target="_self">Th36</a></i>;<br/><b>hence </b><a NAME="E4:48_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="symsp_1.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:48_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:48"><i><font color="Green">E44</font></i></a>, <a class="txt" href="symsp_1.html#E1:48_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="symsp_1.html#E4:48_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="symsp_1.html#T36" target="_self">Th36</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:48_1"><i><font color="Green">E48</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:48"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>8</sub></font> implies  <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:48_2_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E62"><i><font color="Green">Lemma41</font></i></a></i>;<br/><b>hence </b><a NAME="E3:48_2_1"/>
 <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48"><i><font color="Green">E43</font></i></a>, <a class="txt" href="symsp_1.html#E2:48"><i><font color="Green">E44</font></i></a>, <a class="ref" href="symsp_1.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:48_2_2"/><i><font color="Green">E49</font></i>: 
( not <font color="Maroon">c<sub>11</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>7</sub></font> &amp; not <font color="Maroon">c<sub>11</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E3:48"><i><font color="Green">E45</font></i></a>, <a class="ref" href="symsp_1.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E4:48_2_2"/>
 <a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="symsp_1.html#E3:48_2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="symsp_1.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E5:48_2_2"/><b>then </b>
<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">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>11</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>;<br/><b>hence </b><a NAME="E6:48_2_2"/>
 <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:48"><i><font color="Green">E43</font></i></a>, <a class="txt" href="symsp_1.html#E2:48"><i><font color="Green">E44</font></i></a>, <a class="txt" href="symsp_1.html#E3:48_2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="symsp_1.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48_2"/>
 <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:48_2"><i><font color="Green">E48</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:48"/>
(  <a href="symsp_1.html#K2" target="_self">PProJ</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>9</sub></font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E4:48"><i><font color="Green">E46</font></i></a></i>;<br/>


</div>
