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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b3">p</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b3">p</font> is  <a href="matrix16.html#V4" title="MATRIX16:attr.4">first-col-of-circulant</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Olive" title="b3">p</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Olive" title="b3">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Olive" title="b3">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">p</font> is  <a href="matrix16.html#V4" title="MATRIX16:attr.4">first-col-of-circulant</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Olive" title="b1">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> is  <a href="matrix16.html#V4" title="MATRIX16:attr.4">first-col-of-circulant</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c4">p</font></span>)</span> )</span><br/>







<a NAME="E1:63"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>;<br/>
<a NAME="E2:63"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c4">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:63"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c4">p</font> is  <a href="matrix16.html#V4" title="MATRIX16:attr.4">first-col-of-circulant</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c4">p</font></span>)</span></span><br/>

<a NAME="E4:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A4</font></span>: 
( <font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c4">p</font> is  <a href="matrix16.html#V4" title="MATRIX16:attr.4">first-col-of-circulant</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c4">p</font> is  <a href="matrix16.html#V4" title="MATRIX16:attr.4">first-col-of-circulant</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix16.html#T46" target="_self" title="MATRIX16:th.46">Th46</a></span>;<br/>
<a NAME="E5:63"/>
( <font color="Maroon" title="c2">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c4">p</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c4">p</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="matrix16.html#T47" target="_self" title="MATRIX16:th.47">Th47</a></span>;<br/>
<a NAME="E6:63"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E1:63"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="matrix16.html#T38" target="_self" title="MATRIX16:th.38">Th38</a>, <a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:63"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="matrix_6.html#K2" title="MATRIX_6:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K2" title="MATRIX16:func.2">CCirc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="fvsum_1.html#T55" title="FVSUM_1:th.55">FVSUM_1:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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