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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">W</font>, <font color="Maroon" title="c3">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>;<br/><b>let </b><font color="Maroon" title="c4">T</font> be   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c2">W</font>;<br/><b>let </b><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">V</font>;<br/><b>let </b><font color="Maroon" title="c6">B</font> be    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c3">V</font>;<br/><b>let </b><font color="Maroon" title="c7">l</font> be    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">A</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c5">A</font> is    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c4">T</font>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">G</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c3">V</font><b> such that </b><br/><a NAME="E4:54"/><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c8">G</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:54"/><i><font color="Green" title="E43">A4</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">l</font>
 <b>and </b><br/><a NAME="E6:54"/><i><font color="Green" title="E44">A5</font></i>: 
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">l</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c8">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_6.html#D9" title="VECTSP_6:def.9">VECTSP_6:def 9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">H</font> = <font color="Maroon" title="c4">T</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c8">G</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">H</font> = <font color="Maroon" title="c4">T</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c8">G</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">W</font> ;<br/>
<a NAME="E8:54"/><i><font color="Green" title="E45">A6</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">A</font></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="ranknull.html#E1:54"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="ranknull.html#E2:54"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="ranknull.html#T22" target="_self" title="RANKNULL:th.22">Th22</a></i>;<br/>
<a NAME="E9:54"/><i><font color="Green" title="E46">A7</font></i>: 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#D7" title="VECTSP_6:def.7">VECTSP_6:def 7</a></i>;<br/>
<a NAME="E10:54"/><i><font color="Green" title="E47">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ranknull.html#E9:54"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>;<br/>
<a NAME="E11:54"/><i><font color="Green" title="E48">A9</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">l</font></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="ranknull.html#E8:54"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="ranknull.html#E10:54"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/>
<a NAME="E12:54"/><i><font color="Green" title="E49">A10</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c3">V</font>
 
<b>by </b><i><a class="ref" href="ranknull.html#T7" target="_self" title="RANKNULL:th.7">Th7</a></i>;<br/>
<a NAME="E13:54"/><i><font color="Green" title="E50">A11</font></i>: 
<font color="Maroon" title="c10">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="ranknull.html#E4:54"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="ranknull.html#E5:54"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="ranknull.html#E11:54"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="ranknull.html#E12:54"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="ranknull.html#T1" target="_self" title="RANKNULL:th.1">Th1</a></i>;<br/>
<i><font color="Green" title="E51">A12</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">H</font> = 
<font color="Maroon" title="c4">T</font> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c7">l</font></span>)</span>
<b>by </b><i><a class="txt" href="ranknull.html#E5:54"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="relat_1.html#T160" title="RELAT_1:th.160">RELAT_1:160</a></i>
<br/>.= 
 <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c7">l</font></span>)</span>
<b>by </b><i><a class="txt" href="ranknull.html#E11:54"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="ranknull.html#T39" target="_self" title="RANKNULL:th.39">Th39</a></i>
;<br/>
<a NAME="E15:54"/><i><font color="Green" title="E52">A13</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">l</font> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c7">l</font></span>)</span> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c10">H</font>
 
<b>by </b><i><a class="txt" href="ranknull.html#E5:54"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="ranknull.html#E11:54"><i><font color="Green" title="E48">A9</font></i></a>, <a class="ref" href="ranknull.html#T38" target="_self" title="RANKNULL:th.38">Th38</a></i>;<br/>
<a NAME="E16:54"/>
 <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c7">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c7">l</font></span>)</span> <a href="vectsp_6.html#K3" title="VECTSP_6:func.3">(#)</a> <font color="Maroon" title="c10">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ranknull.html#E13:54"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="ranknull.html#E14:54"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="vectsp_6.html#D9" title="VECTSP_6:def.9">VECTSP_6:def 9</a></i>;<br/>
<b>hence </b><a NAME="E17:54"/>
<font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c7">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ranknull.html#K8" title="RANKNULL:func.8">@</a> <font color="Maroon" title="c7">l</font></span>)</span>
 <b>by </b><i><a class="txt" href="ranknull.html#E6:54"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="ranknull.html#E15:54"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="matrlin.html#T20" title="MATRLIN:th.20">MATRLIN:20</a></i>;<br/>


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