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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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>, <font color="Olive" title="b4">q</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#V2" title="MATRIX16:attr.2">first-line-of-circulant</a> &amp; <font color="Olive" title="b4">q</font> is <a href="matrix16.html#V2" title="MATRIX16:attr.2">first-line-of-circulant</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font> <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="Olive" title="b4">q</font> 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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Olive" title="b3">p</font></span>)</span></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Olive" title="b4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K1" title="MATRIX16:func.1">LCirc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Olive" title="b3">p</font></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <font color="Olive" title="b4">q</font></span>)</span></span>)</span></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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#V2" title="MATRIX16:attr.2">first-line-of-circulant</a> &amp; <font color="Olive" title="b2">q</font> is <a href="matrix16.html#V2" title="MATRIX16:attr.2">first-line-of-circulant</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <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="Olive" title="b2">q</font> 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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Olive" title="b2">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K1" title="MATRIX16:func.1">LCirc</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="Olive" title="b1">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="Olive" title="b2">q</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> is <a href="matrix16.html#V2" title="MATRIX16:attr.2">first-line-of-circulant</a> &amp; <font color="Maroon" title="c5">q</font> is <a href="matrix16.html#V2" title="MATRIX16:attr.2">first-line-of-circulant</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <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="c5">q</font> 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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K1" title="MATRIX16:func.1">LCirc</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="c5">q</font></span>)</span></span>)</span> )</span><br/>









<b>set </b><font color="Maroon" title="c6">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font>;<br/>
<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c4">p</font> is <a href="matrix16.html#V2" title="MATRIX16:attr.2">first-line-of-circulant</a> &amp; <font color="Maroon" title="c5">q</font> is <a href="matrix16.html#V2" title="MATRIX16:attr.2">first-line-of-circulant</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <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="c5">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K1" title="MATRIX16:func.1">LCirc</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="c5">q</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:60"/><b>then </b><i><font color="Green" title="E31">A4</font></i>: 
( <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#V2" title="MATRIX16:attr.2">first-line-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="c5">q</font> is <a href="matrix16.html#V2" title="MATRIX16:attr.2">first-line-of-circulant</a> )
 
<b>by </b><i><a class="ref" href="matrix16.html#T41" target="_self" title="MATRIX16:th.41">Th41</a></i>;<br/>
<a NAME="E3:60"/><i><font color="Green" title="E32">A6</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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> <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> &amp;  <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="c5">q</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> )
 
<b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>;<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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix16.html#K1" title="MATRIX16:func.1">LCirc</a> <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></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="matrix16.html#T42" target="_self" title="MATRIX16:th.42">Th42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix16.html#K1" title="MATRIX16:func.1">LCirc</a> <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></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix16.html#K1" title="MATRIX16:func.1">LCirc</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="c5">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="matrix16.html#T42" target="_self" title="MATRIX16:th.42">Th42</a></i>
<br/>.= 
 <a href="matrix16.html#K1" title="MATRIX16:func.1">LCirc</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="c5">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:60"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="#E3:60"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="matrix16.html#T34" target="_self" title="MATRIX16:th.34">Th34</a></i>
;<br/>
<b>hence </b><a NAME="E5:60"/>
<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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</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#K1" title="MATRIX16:func.1">LCirc</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix16.html#K1" title="MATRIX16:func.1">LCirc</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="c5">q</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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