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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b4">M1</font>, <font color="Olive" title="b5">M2</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font>  st <font color="Olive" title="b4">M1</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> &amp; <font color="Olive" title="b5">M2</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Olive" title="b4">M1</font></span>)</span> <a href="matrix_6.html#K3" title="MATRIX_6:func.3">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Olive" title="b5">M2</font></span>)</span> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a></span><br/><b>let </b><font color="Maroon" title="c2">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="c2">K</font><br/> for <font color="Olive" title="b3">M1</font>, <font color="Olive" title="b4">M2</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b3">M1</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> &amp; <font color="Olive" title="b4">M2</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_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> <font color="Olive" title="b3">M1</font></span>)</span> <a href="matrix_6.html#K3" title="MATRIX_6: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> <font color="Olive" title="b4">M2</font></span>)</span> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a></span><br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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">M1</font>, <font color="Olive" title="b2">M2</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">M1</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> &amp; <font color="Olive" title="b2">M2</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Olive" title="b1">M1</font></span>)</span> <a href="matrix_6.html#K3" title="MATRIX_6:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Olive" title="b2">M2</font></span>)</span> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a></span><br/><b>let </b><font color="Maroon" title="c5">M1</font>, <font color="Maroon" title="c6">M2</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">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="c5">M1</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> &amp; <font color="Maroon" title="c6">M2</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_6.html#K3" title="MATRIX_6:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> )</span><br/>











<b>assume </b><a NAME="E1:28"/>
( <font color="Maroon" title="c5">M1</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> &amp; <font color="Maroon" title="c6">M2</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> )
 ; <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="c3">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_6.html#K3" title="MATRIX_6:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a></span><br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">M1</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> &amp; <font color="Maroon" title="c4">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">M2</font> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> )
 
<b>by </b><i><a class="ref" href="matrix16.html#T6" target="_self" title="MATRIX16:th.6">Th6</a></i>;<br/>
<a NAME="E3:28"/><b>then </b>
 <a href="matrix_6.html#K1" title="MATRIX_6:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a>
 
<b>by </b><i><a class="ref" href="matrix16.html#T11" target="_self" title="MATRIX16:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E4:28"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_6.html#K3" title="MATRIX_6:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> is <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a>
 <b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="matrix16.html#T7" target="_self" title="MATRIX16:th.7">Th7</a></i>; <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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