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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#NK1" title="NUMBERS:NK.1">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K2" title="POLYNOM8:func.2">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#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Olive" title="b5">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0: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#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b2">s</font></span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#NK1" title="NUMBERS:NK.1">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K2" title="POLYNOM8:func.2">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#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Olive" title="b4">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0: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#K4" title="RLVECT_1:func.4">Sum</a> <font color="Olive" title="b1">s</font></span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#NK1" title="NUMBERS:NK.1">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K2" title="POLYNOM8:func.2">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#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Olive" title="b3">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0: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#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">s</font></span><br/><span class="kw">let </span><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#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K2" title="POLYNOM8:func.2">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#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0: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#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">s</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:60"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:60"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
( 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> )
 <span class="kw">and </span><br/><a NAME="E3:60"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 <span class="kw">and </span><br/><a NAME="E4:60"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:60"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E6:60"/><span class="lab"><font color="Green" title="E59">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K2" title="POLYNOM8:func.2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0: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#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">s</font></span><br/>

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>,<font color="Maroon" title="c5">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E60">A7</font></span>:  <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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#T6" onmouseover="rs('matrix_3/T6')" onmouseout="rh()">MATRIX_3:6</a></span>
<br/>.= 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/>
<a NAME="E9:60"/><span class="lab"><font color="Green" title="E61">A8</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="polynom8.html#T30" target="_self" onmouseover="rs('polynom8/T30')" onmouseout="rh()">Th30</a></span>;<br/>
<a NAME="E10:60"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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#K2" title="POLYNOM8:func.2">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_3"><span class="kw">proof </span></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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/>
<a NAME="E2:60_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A10</font></span>: 
 <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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>,<font color="Maroon" title="c5">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/>
<a NAME="E4:60_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A11</font></span>: 
 <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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:60_3"/><span class="lab"><font color="Green" title="E65">A12</font></span>: 
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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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#K2" title="POLYNOM8:func.2">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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:60_3"/><span class="lab"><font color="Green" title="E66">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">and </span><br/><a NAME="E7:60_3"/><span class="lab"><font color="Green" title="E67">A14</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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#K2" title="POLYNOM8:func.2">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="E8:60_3"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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> &amp; <font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_3"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E7:60_3"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>;<br/>
<a NAME="E9:60_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A15</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI: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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:60_3"/>
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_3"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E7:60_3"><span class="lab"><font color="Green" title="E67">A14</font></span></a></span>;<br/>
<a NAME="E11:60_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A16</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_3"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E12:60_3"/>
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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_3"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<a NAME="E13:60_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A17</font></span>: 
<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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_3"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<a NAME="E14:60_3"/>
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>;<br/>
<a NAME="E15:60_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI: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_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_3"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E7:60_3"><span class="lab"><font color="Green" title="E67">A14</font></span></a></span>;<br/>
<a NAME="E16:60_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A18</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c7">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/>
<a NAME="E17:60_3"/>
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_3"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E7:60_3"><span class="lab"><font color="Green" title="E67">A14</font></span></a></span>;<br/>
<a NAME="E18:60_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A19</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_3"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c5">j</font>) = 
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E4:60"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E6:60_3"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E7:60_3"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="ref" href="polynom8.html#D8" target="_self" onmouseover="rs('polynom8/D8')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E13:60_3"><span class="lab"><font color="Green" title="E70">A17</font></span></a>, <a class="ref" href="polynom8.html#T22" target="_self" onmouseover="rs('polynom8/T22')" onmouseout="rh()">Th22</a></span>
;<br/>
<a NAME="E20:60_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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#K2" title="POLYNOM8:func.2">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_3"><span class="lab"><font color="Green" title="E68">A15</font></span></a>, <a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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#K2" title="POLYNOM8:func.2">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#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E6:60_3"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E7:60_3"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="txt" href="#E11:60_3"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="txt" href="#E16:60_3"><span class="lab"><font color="Green" title="E71">A18</font></span></a>, <a class="txt" href="#E18:60_3"><span class="lab"><font color="Green" title="E72">A19</font></span></a>, <a class="ref" href="polynom8.html#D8" target="_self" onmouseover="rs('polynom8/D8')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="ref" href="polynom8.html#T23" target="_self" onmouseover="rs('polynom8/T23')" onmouseout="rh()">Th23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E22:60_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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#K2" title="POLYNOM8:func.2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:60"/><span class="lab"><font color="Green" title="E63">A20</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_4"><span class="kw">proof </span></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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/>
<a NAME="E2:60_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A21</font></span>: 
 <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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>,<font color="Maroon" title="c5">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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 href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c6">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c6">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/>
<a NAME="E4:60_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A22</font></span>: 
 <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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:60_4"/><span class="lab"><font color="Green" title="E66">A23</font></span>: 
( 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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E67">A24</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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#K2" title="POLYNOM8:func.2">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60"><span class="lab"><font color="Green" title="E62">A9</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E5:60_4"><span class="lab"><font color="Green" title="E66">A23</font></span></a></span>
;<br/>
<a NAME="E7:60_4"/>
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_4"><span class="lab"><font color="Green" title="E66">A23</font></span></a></span>;<br/>
<a NAME="E8:60_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A25</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_4"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E9:60_4"/>
 <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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>,<font color="Maroon" title="c5">j</font>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:60"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:60_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A26</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI: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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>,<font color="Maroon" title="c5">j</font>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_4"><span class="lab"><font color="Green" title="E66">A23</font></span></a></span>;<br/>
<a NAME="E11:60_4"/>
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_4"><span class="lab"><font color="Green" title="E66">A23</font></span></a></span>;<br/>
<a NAME="E12:60_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_4"><span class="lab"><font color="Green" title="E65">A22</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E13:60_4"/><span class="kw">then </span>
<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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_4"><span class="lab"><font color="Green" title="E69">A26</font></span></a>, <a class="txt" href="#E8:60_4"><span class="lab"><font color="Green" title="E68">A25</font></span></a>, <a class="ref" href="fvsum_1.html#T60" onmouseover="rs('fvsum_1/T60')" onmouseout="rh()">FVSUM_1:60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:60_4"/>
<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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_4"><span class="lab"><font color="Green" title="E69">A26</font></span></a>, <a class="txt" href="#E6:60_4"><span class="lab"><font color="Green" title="E67">A24</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:60"/><span class="lab"><font color="Green" title="E64">A27</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI: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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI: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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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/>

<span class="kw">assume </span><a NAME="E1:60_5"/><span class="lab"><font color="Green" title="E65">A28</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI: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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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"/><span class="lab"><font color="Green" title="E66">A29</font></span>: 
 <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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>,<font color="Maroon" title="c5">j</font>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:60"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:60_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A30</font></span>: 
( 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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_5"><span class="lab"><font color="Green" title="E65">A28</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:60_5"/><span class="lab"><font color="Green" title="E68">A31</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="#E1:60_5"><span class="lab"><font color="Green" title="E65">A28</font></span></a>, <a class="txt" href="#E2:60_5"><span class="lab"><font color="Green" title="E66">A29</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_5"><span class="lab"><font color="Green" title="E65">A28</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:60"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="txt" href="#E3:60_5"><span class="lab"><font color="Green" title="E67">A30</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_5"><span class="lab"><font color="Green" title="E68">A31</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:60_5"/>
<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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:60"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E14:60"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A32</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:60"><span class="lab"><font color="Green" title="E64">A27</font></span></a>, <a class="ref" href="finseq_1.html#T13" onmouseover="rs('finseq_1/T13')" onmouseout="rh()">FINSEQ_1:13</a></span>;<br/>
<a NAME="E15:60"/>
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>;<br/>
<a NAME="E16:60"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A33</font></span>: 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>;<br/>
<a NAME="E17:60"/><span class="lab"><font color="Green" title="E67">A34</font></span>: 
(  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>;<br/>
<a NAME="E18:60"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>;<br/>
<a NAME="E19:60"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A35</font></span>: 
 <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#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:60"><span class="lab"><font color="Green" title="E67">A34</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>;<br/>
<a NAME="E20:60"/>
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>;<br/>
<a NAME="E21:60"/><span class="kw">then </span>
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:60"><span class="lab"><font color="Green" title="E67">A34</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>;<br/>
<a NAME="E22:60"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span> is   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E19:60"><span class="lab"><font color="Green" title="E68">A35</font></span></a>, <a class="ref" href="matrix_0.html#T20" onmouseover="rs('matrix_0/T20')" onmouseout="rh()">MATRIX_0:20</a></span>;<br/>
<a NAME="E23:60"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A36</font></span>: 
 <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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#K8" title="MCART_1:func.8">[:</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#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>;<br/>
<a NAME="E24:60"/>
( <font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c5">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E4:60"><span class="lab"><font color="Green" title="E57">A4</font></span></a></span>;<br/>
<a NAME="E25:60"/><span class="kw">then </span>
<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_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:60"><span class="lab"><font color="Green" title="E69">A36</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E26:60"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0: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_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span>,<font color="Maroon" title="c5">j</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:60"><span class="lab"><font color="Green" title="E66">A33</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:60"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<font color="Maroon" title="c6">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0: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#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:60"><span class="lab"><font color="Green" title="E65">A32</font></span></a>, <a class="ref" href="fvsum_1.html#D9" onmouseover="rs('fvsum_1/D9')" onmouseout="rh()">FVSUM_1:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
