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<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_2.html#NM7">RightMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_2.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">l</font> be    <a href="rmod_4.html#M2" target="_self">Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>;<br/>







<a NAME="E1:39"/><i><font color="Green">E48</font></i>: 
 <a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">l</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1"/>
(  <a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1"/>
 <a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:39_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">l</font> <a href="funct_2.html#R2">=</a>  <a href="rmod_4.html#K3" target="_self">ZeroLC</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>hence </b> <a href="rmod_4.html#K5" target="_self">Sum</a> <font color="Maroon">l</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_1.html#T37">MOD_1:37</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_2"/>
 <a href="rmod_4.html#K2" target="_self">Carrier</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:39_1_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>and </b><br/><a NAME="E4:39_1_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="rmod_4.html#K5" target="_self">Sum</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:39_1_1_2"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T106">FINSEQ_3:106</a></i>;<br/><a NAME="E6:39_1_1_2"/><b>then </b>
<font color="Maroon">l</font> <a href="rmod_4.html#K4" target="_self">(#)</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:39_1_1_2"/>
 <a href="rmod_4.html#K5" target="_self">Sum</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:39"/>
 <a href="rmod_4.html#K5" target="_self">Sum</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span>
 ;<br/>


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