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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#NM6">LeftMod</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font>;<br/>







<b>set </b><font color="Maroon">l</font> = <font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font>;<br/>
<a NAME="E1:4"/>
 <a href="vectsp_6.html#K1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T58">VECTSP_6:58</a></i>;<br/>
<b>then consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:4"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E4:4"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E5:4"/><i><font color="Green">E34</font></i>: 
 <a href="vectsp_6.html#K4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E6:4"/><i><font color="Green">E35</font></i>: 
 <a href="vectsp_6.html#K4">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="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="vectsp_6.html#D9">VECTSP_6:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font>;<br/>
<i><font color="Green">E38</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>
;<br/>
<a NAME="E8:4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D8">VECTSP_6:def 8</a></i>;<br/>
<a NAME="E9:4"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E2:4_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">v'</font> = <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>;<br/><a NAME="E3:4_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="vectsp_6.html#T32">VECTSP_6:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#D12">VECTSP_6:def 12</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font>
<b>by </b><i>, , , , <a class="ref" href="vectsp_6.html#T32">VECTSP_6:32</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:4"/>
 <a href="vectsp_6.html#K4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="vectsp_3.html#T9">VECTSP_3:9</a></i>;<br/>


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