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<span class="kw">let </span><font color="Maroon" title="c1">m</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="c1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  for <font color="Olive" title="b1">L</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</a> <br/>  for <font color="Olive" title="b2">p</font> being   <a href="algseq_1.html#NM1" title="ALGSEQ_1:NM.1">AlgSequence</a> of <font color="Olive" title="b1">L</font> holds <br/> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">i</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:45"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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">L</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</a> <br/>  for <font color="Olive" title="b2">p</font> being   <a href="algseq_1.html#NM1" title="ALGSEQ_1:NM.1">AlgSequence</a> of <font color="Olive" title="b1">L</font> holds <br/> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">i</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">L</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</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">p</font> being   <a href="algseq_1.html#NM1" title="ALGSEQ_1:NM.1">AlgSequence</a> of <font color="Maroon" title="c2">L</font> holds <br/> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">i</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="algseq_1.html#NM1" title="ALGSEQ_1:NM.1">AlgSequence</a> of <font color="Maroon" title="c2">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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font> &amp;  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">q</font> =  <a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>);<br/>
<span class="kw">thus </span><a NAME="E2:45"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</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"> (  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:45"/>
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</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">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">now </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">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</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="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> implies <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:45_1"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">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="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">i</font></span><br/><a NAME="E2:45_1"/><span class="kw">then </span>
( <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:45_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#D4" target="_self" onmouseover="rs('polynom8/D4')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:45_1"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">i</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>
<span class="kw">hence </span><a NAME="E5:45"/>
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</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/>


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