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<b>let </b><font color="Maroon" title="c1">L</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">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font>, <font color="Olive" title="b5">m</font> being   <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>   st <font color="Olive" title="b1">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Olive" title="b5">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">m</font> &amp; ( for <font color="Olive" title="b6">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">k</font> &amp; <font color="Olive" title="b6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">m</font> holds <br/><font color="Olive" title="b2">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> ) holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b5">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Olive" title="b5">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b2">s</font></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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">s</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font>, <font color="Olive" title="b4">m</font> being   <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>   st <font color="Maroon" title="c2">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Olive" title="b4">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">m</font> &amp; ( for <font color="Olive" title="b5">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">k</font> &amp; <font color="Olive" title="b5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/><font color="Olive" title="b1">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> ) holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b4">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Olive" title="b4">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b1">s</font></span><br/><b>let </b><font color="Maroon" title="c3">s</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</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">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">m</font> being   <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>   st <font color="Maroon" title="c2">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Olive" title="b3">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">m</font> &amp; ( for <font color="Olive" title="b4">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><font color="Maroon" title="c3">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> ) holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Olive" title="b3">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c3">s</font></span><br/><b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c6">m</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"> ( <font color="Maroon" title="c2">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Maroon" title="c6">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> holds <br/><font color="Maroon" title="c3">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> ) implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c3">s</font> )</span><br/>







<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E56">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Maroon" title="c6">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> holds <br/><font color="Maroon" title="c3">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> ) )
 ; <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"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c3">s</font></span><br/>

<a NAME="E2:60"/><b>then </b><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="polynom8.html#T30" target="_self" title="POLYNOM8:th.30">Th30</a></i>;<br/>
<a NAME="E3:60"/><i><font color="Green" title="E58">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<a NAME="E1:60_1"/><i><font color="Green" title="E59">A4</font></i>: 
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E2:60_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E3:60_1"/><b>then </b>
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> )
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E4:60_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span> is   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></i>;<br/>
<a NAME="E5:60_1"/><b>then </b><i><font color="Green" title="E60">A5</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E6:60_1"/><i><font color="Green" title="E61">A6</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60"><i><font color="Green" title="E56">A1</font></i></a></i>;<br/>
<a NAME="E7:60_1"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60"><i><font color="Green" title="E56">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:60_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E5:60_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="polynom8.html#E6:60_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:60"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</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> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<a NAME="E1:60_2"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<b>hence </b><a NAME="E2:60_2"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</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> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:60"/><b>then </b><i><font color="Green" title="E59">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:60"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E6:60"/><i><font color="Green" title="E60">A8</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:60_3"/><i><font color="Green" title="E61">A9</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</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"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E2:60_3"/><i><font color="Green" title="E62">A10</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_1"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:60_3_1"/><b>then </b><i><font color="Green" title="E63">A11</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_3_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_3"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:60_3_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E2:60_3_1"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:60_3"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E4:60_3"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_3"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:60_3"/><b>then </b><i><font color="Green" title="E63">A12</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E6:60_3"/><i><font color="Green" title="E64">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:60_3_2"/><i><font color="Green" title="E65">A14</font></i>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_3"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E2:60_3_2"/>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E3:60_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="polynom8.html#E1:60_3_2"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:60_3"/><i><font color="Green" title="E65">A15</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_3"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:60_3_3"/><b>then </b><i><font color="Green" title="E66">A16</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_3_3"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_3"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:60_3_3"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E2:60_3_3"><i><font color="Green" title="E66">A16</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:60_3"/><i><font color="Green" title="E66">A17</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c5">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_4"><b>proof </b></a><div class="add">

<a NAME="E1:60_3_4"/><i><font color="Green" title="E67">A18</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E2:60_3_4"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_3"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:60_3_4"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_3_4"><i><font color="Green" title="E67">A18</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:60_3_4"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:60_3"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_5"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c5">j</font> = 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:60"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="polynom8.html#E1:60_3"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="polynom8.html#D8" target="_self" title="POLYNOM8:def.8">Def8</a></i>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:60"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="polynom8.html#E6:60_3"><i><font color="Green" title="E64">A13</font></i></a>, <a class="ref" href="polynom8.html#T22" target="_self" title="POLYNOM8:th.22">Th22</a></i>
;<br/>
<b>hence </b><a NAME="E2:60_3_5"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E8:60_3"><i><font color="Green" title="E66">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:60"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="polynom8.html#E1:60_3"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="polynom8.html#E2:60_3"><i><font color="Green" title="E62">A10</font></i></a>, <a class="txt" href="polynom8.html#E5:60_3"><i><font color="Green" title="E63">A12</font></i></a>, <a class="txt" href="polynom8.html#E7:60_3"><i><font color="Green" title="E65">A15</font></i></a>, <a class="ref" href="polynom8.html#D8" target="_self" title="POLYNOM8:def.8">Def8</a></i>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:60"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="polynom8.html#T23" target="_self" title="POLYNOM8:th.23">Th23</a></i>
;<br/>
<b>hence </b><a NAME="E11:60_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</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/>


</div><b>end;</b></div>
<a NAME="E7:60"/><i><font color="Green" title="E61">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_4"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
;<br/>
<b>then </b> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T8" title="MATRIX_3:th.8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<b>hence </b><a NAME="E3:60_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:60"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c3">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5"><b>proof </b></a><div class="add">

<a NAME="E1:60_5"/><i><font color="Green" title="E62">A20</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> )</span><br/>

<b>assume </b><a NAME="E1:60_5_1"/><i><font color="Green" title="E63">A21</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</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"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span><br/>

<a NAME="E2:60_5_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E7:60"><i><font color="Green" title="E61">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_5_1"/><b>then </b><i><font color="Green" title="E64">A22</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_5_1"><i><font color="Green" title="E63">A21</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:60_5_1"/><i><font color="Green" title="E65">A23</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:60_5_1_1"/><i><font color="Green" title="E66">A24</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_1_1_1"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:60_5_1_1_1"/><b>then </b><i><font color="Green" title="E67">A25</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_5_1_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_5_1"><i><font color="Green" title="E63">A21</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:60_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E2:60_5_1_1_1"><i><font color="Green" title="E67">A25</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:60_5_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_1_1_2"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:60_5_1_1_2"/><b>then </b><i><font color="Green" title="E67">A26</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_5_1_1_2"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_5_1"><i><font color="Green" title="E63">A21</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:60_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E2:60_5_1_1_2"><i><font color="Green" title="E67">A26</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_5_1_1"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E3:60_5_1"><i><font color="Green" title="E64">A22</font></i></a>, <a class="txt" href="polynom8.html#E1:60_5_1_1"><i><font color="Green" title="E66">A24</font></i></a>, <a class="ref" href="fvsum_1.html#T73" title="FVSUM_1:th.73">FVSUM_1:73</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font></span>)</span> = 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E6:60"><i><font color="Green" title="E60">A8</font></i></a>, <a class="txt" href="polynom8.html#E1:60_5_1"><i><font color="Green" title="E63">A21</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:60"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="polynom8.html#E1:60_5_1"><i><font color="Green" title="E63">A21</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:60_5_1"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E3:60_5_1"><i><font color="Green" title="E64">A22</font></i></a>, <a class="txt" href="polynom8.html#E4:60_5_1"><i><font color="Green" title="E65">A23</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:60_5"/>
 <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_2"><b>proof </b></a><div class="add">

<i><font color="Green" title="E63">A27</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="txt" href="polynom8.html#E7:60"><i><font color="Green" title="E61">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:60"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E2:60_5_2"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> )</span><br/>

<b>assume </b><a NAME="E1:60_5_2_2"/><i><font color="Green" title="E64">A28</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span>
 ; <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"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font></span><br/>

<a NAME="E2:60_5_2_2"/><i><font color="Green" title="E65">A29</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E7:60"><i><font color="Green" title="E61">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_5_2_2"/><b>then </b><i><font color="Green" title="E66">A30</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> )
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_5_2_2"><i><font color="Green" title="E64">A28</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:60_5_2_2"/><i><font color="Green" title="E67">A31</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:60"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="polynom8.html#E1:60_5_2_2"><i><font color="Green" title="E64">A28</font></i></a>, <a class="txt" href="polynom8.html#E2:60_5_2_2"><i><font color="Green" title="E65">A29</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> = 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_5_2_2"><i><font color="Green" title="E64">A28</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c3">s</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:60_5"><i><font color="Green" title="E62">A20</font></i></a>, <a class="txt" href="polynom8.html#E3:60_5_2_2"><i><font color="Green" title="E66">A30</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="polynom8.html#E4:60_5_2_2"><i><font color="Green" title="E67">A31</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E6:60_5_2_2"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_5_2"/>
 <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:60_5_2"><i><font color="Green" title="E63">A27</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_5"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c3">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:60"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>,<font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E5:60"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="fvsum_1.html#D10" title="FVSUM_1:def.10">FVSUM_1:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
