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<b>let </b><font color="Maroon" title="c4">F</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c5">S</font> be   <a href="symsp_1.html#NM1" title="SYMSP_1:NM.1">SymSp</a> of <font color="Maroon" title="c4">F</font>;<br/><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font>;<br/>









<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E18">A1</font></i>: 
( <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c5">S</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c5">S</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c5">S</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font><b> such that </b><br/><a NAME="E3:22"/><i><font color="Green" title="E19">A2</font></i>: 
( not <font color="Maroon" title="c9">r</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c6">a</font> &amp; not <font color="Maroon" title="c9">r</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="ref" href="symsp_1.html#T21" target="_self" title="SYMSP_1:th.21">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font><b> such that </b><br/><a NAME="E5:22"/><i><font color="Green" title="E20">A3</font></i>: 
( not <font color="Maroon" title="c10">s</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c6">a</font> &amp; not <font color="Maroon" title="c10">s</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="symsp_1.html#T21" target="_self" title="SYMSP_1:th.21">Th21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:22_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c9">r</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">c</font>
 <b>and </b><br/><a NAME="E2:22_1"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c10">s</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">b</font>
 ;<br/><div><i><font color="Green" title="E23">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1_1"/><i><font color="Green" title="E24">A7</font></i>: 
not <font color="Maroon" title="c9">r</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">s</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c6">a</font>
 ;<br/><a NAME="E2:22_1_1"/>
( not <font color="Maroon" title="c9">r</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">s</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">b</font> &amp; not <font color="Maroon" title="c9">r</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">s</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E3:22"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="symsp_1.html#E5:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="symsp_1.html#E1:22_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="symsp_1.html#E2:22_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="symsp_1.html#T14" target="_self" title="SYMSP_1:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E3:22_1_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c6">a</font> &amp; not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">b</font> &amp; not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:22_1_1"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1_2"/><i><font color="Green" title="E24">A8</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">s</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c6">a</font>
 ;<br/><a NAME="E2:22_1_2"/>
( not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">s</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">b</font> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">s</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">c</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:22_1_2_1"/>
( not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">r</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">b</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">r</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">c</font> )
 
<b>by </b><i><a class="txt" href="symsp_1.html#E1:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="symsp_1.html#E3:22"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="symsp_1.html#E1:22_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="symsp_1.html#D2" target="_self" title="SYMSP_1:def.2">Def2</a>, <a class="ref" href="symsp_1.html#T15" target="_self" title="SYMSP_1:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E2:22_1_2_1"/>
( not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">s</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">b</font> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c4">F</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c10">s</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E5:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="symsp_1.html#E2:22_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="symsp_1.html#T14" target="_self" title="SYMSP_1:th.14">Th14</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:22_1_2"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c6">a</font> &amp; not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">b</font> &amp; not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:22_1_2"><i><font color="Green" title="E24">A8</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:22_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c6">a</font> &amp; not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">b</font> &amp; not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E3:22"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="symsp_1.html#E3:22_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="symsp_1.html#T19" target="_self" title="SYMSP_1:th.19">Th19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:22"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S</font> st <br/>( not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c6">a</font> &amp; not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">b</font> &amp; not <font color="Olive" title="b1">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">c</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E3:22"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="symsp_1.html#E5:22"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>


</div>
