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<b>let </b><font color="Maroon" title="c4">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">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">b</font>, <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c10">z</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><b>that </b><br/><a NAME="E1:51"/><i><font color="Green" title="E43">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#K3" title="RLVECT_1:func.3">+</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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green" title="E44">A2</font></i>: 
not <font color="Maroon" title="c6">b</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">a</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c11">0F</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font>;<br/>
<div><i><font color="Green" title="E45">A3</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" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font>
 ;<br/><a NAME="E2:51_1"/><b>then </b>
(  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> &amp;  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> &amp;  <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">F</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:51"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="symsp_1.html#E2:51"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="symsp_1.html#T47" target="_self" title="SYMSP_1:th.47">Th47</a></i>;<br/><b>hence </b><a NAME="E3:51_1"/>
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">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" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font>
 ;<br/><a NAME="E2:51_2"/><b>then </b>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">S</font> )
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:51"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="symsp_1.html#E62"><i><font color="Green" title="E41">Lm6</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">p</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="E4:51_2"/><i><font color="Green" title="E46">A4</font></i>: 
( not <font color="Maroon" title="c12">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c7">a</font> &amp; not <font color="Maroon" title="c12">p</font> <a href="symsp_1.html#R1" title="SYMSP_1:pred.1">_|_</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="ref" href="symsp_1.html#T21" target="_self" title="SYMSP_1:th.21">Th21</a></i>;<br/><a NAME="E5:51_2"/><i><font color="Green" title="E47">A5</font></i>: 
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">p</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:51"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="symsp_1.html#E2:51"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="symsp_1.html#E4:51_2"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="symsp_1.html#D7" target="_self" title="SYMSP_1:def.7">Def7</a></i>;<br/><a NAME="E6:51_2"/><i><font color="Green" title="E48">A6</font></i>: 
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">p</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c9">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:51"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="symsp_1.html#E2:51"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="symsp_1.html#E4:51_2"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="symsp_1.html#D7" target="_self" title="SYMSP_1:def.7">Def7</a></i>;<br/><a NAME="E7:51_2"/><i><font color="Green" title="E49">A7</font></i>: 
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c12">p</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c10">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="symsp_1.html#E1:51"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="symsp_1.html#E2:51"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="symsp_1.html#E4:51_2"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="symsp_1.html#D7" target="_self" title="SYMSP_1:def.7">Def7</a></i>;<br/><a NAME="E8:51_2"/>
 <a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K1" title="SYMSP_1:func.1">ProJ</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c10">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="symsp_1.html#E4:51_2"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="symsp_1.html#T29" target="_self" title="SYMSP_1:th.29">Th29</a></i>;<br/><b>hence </b><a NAME="E9:51_2"/>
 <a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="symsp_1.html#E5:51_2"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="symsp_1.html#E6:51_2"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="symsp_1.html#E7:51_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">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" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="symsp_1.html#K2" title="SYMSP_1:func.2">PProJ</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="symsp_1.html#E3:51"><i><font color="Green" title="E45">A3</font></i></a></i>;<br/>


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