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<b>let </b><font color="Maroon">GF</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:5"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<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>assume </b><a NAME="E2:5"/><i><font color="Green">E40</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>consider </b><font color="Maroon">G</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="E4:5"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">G</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:5"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</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="E6:5"/><i><font color="Green">E44</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">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9">VECTSP_6:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font><b> such that </b><br/><a NAME="E9:5"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T124">FINSEQ_3:124</a></i>;<br/>
<b>set </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>;<br/>
<a NAME="E10:5"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">X</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> ;<br/>
<font color="Maroon">X</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E13:5"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> &amp; <font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a>, <a class="ref" href="finseq_3.html#T94">FINSEQ_3:94</a></i>;<br/>
<a NAME="E14:5"/><b>then </b>
<font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>,<font color="Maroon">G</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T39">RFINSEQ:39</a></i>;<br/>
<b>then consider </b><font color="Maroon">Q</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E16:5"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="rfinseq.html#T17">RFINSEQ:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>, <font color="Maroon">F2</font> = <font color="Maroon">F</font> <a href="finseq_3.html#K1">-</a> <font color="Maroon">A</font> as    <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>by </b><i><a class="ref" href="finseq_3.html#T93">FINSEQ_3:93</a></i>;<br/>
<a NAME="E18:5"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>;<br/>
<a NAME="E19:5"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F2</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
 <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> = 
 <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> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F2</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T37">VECTSP_6:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E21:5"/>
 <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> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">L</font>
 ;<br/>


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