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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>set </b><font color="Maroon">0F</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">0V</font> =  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>;<br/>
<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1"/><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> &amp; <font color="Maroon">x</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font> )
 ;<br/><a NAME="E2:36_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</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">x</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E3:36_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="symsp_1.html#E2:36_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/><a NAME="E4:36_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>;<br/><a NAME="E5:36_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>;<br/><a NAME="E6:36_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>;<br/><b>hence </b><a NAME="E7:36_1"/>
 <a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</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#E2:36_1"><i><font color="Green">E32</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_2"/>
( not <font color="Maroon">b</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font> &amp;  <a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> )
 ;<br/><a NAME="E2:36_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">b</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E3:36_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T59">VECTSP_1:59</a></i>;<br/><a NAME="E4:36_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>;<br/><a NAME="E5:36_2"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T25">RLVECT_1:25</a></i>;<br/><b>hence </b><a NAME="E6:36_2"/>
<font color="Maroon">x</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:36"/>
( not <font color="Maroon">b</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font> implies ( <font color="Maroon">x</font> <a href="symsp_1.html#R1" target="_self">_|_</a> <font color="Maroon">a</font> iff  <a href="symsp_1.html#K1" target="_self">ProJ</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">F</font> ) )
 <b>by </b><i/>;<br/>


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