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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</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="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Olive" title="b1">A</font> holds <br/> <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Olive" title="b2">B</font></span>)</span>,<font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font>)</span>)</span> <a href="matrlin2.html#K5" title="MATRLIN2:func.5">+</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</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="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>, <font color="Olive" title="b2">Q</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">A</font> holds <br/> <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c3">B</font></span>)</span>,<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>)</span>)</span> <a href="matrlin2.html#K5" title="MATRLIN2:func.5">+</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">Q</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">A</font> implies  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="matrlin2.html#K5" title="MATRLIN2:func.5">+</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> )</span><br/>





<a NAME="E1:3"/>
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D14" onmouseover="rs('finseq_1/D14')" onmouseout="rh()">FINSEQ_1:def 14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:3"/>
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">A</font> implies  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> <a href="matrlin2.html#K5" title="MATRLIN2:func.5">+</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp11.html#T1" target="_self" onmouseover="rs('vectsp11/T1')" onmouseout="rh()">Th1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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