<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</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">S</font> being   <a href="field_4.html#NM1" title="FIELD_4:NM.1">RingExtension</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> holds <br/> <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Olive" title="b2">a</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Olive" title="b3">b</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="field_4.html#NM1" title="FIELD_4:NM.1">RingExtension</a> of <font color="Maroon" title="c1">R</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">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/> <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Olive" title="b1">a</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Olive" title="b2">b</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</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">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/> <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Olive" title="b1">b</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</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"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>) )</span><br/>







<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</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="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span><br/>

<a NAME="E2:42"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c1">R</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D1" target="_self" onmouseover="rs('field_4/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E3:42"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T10" target="_self" onmouseover="rs('field_4/T10')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">q</font> =  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">p</font> =  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>) as    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">R</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D10" onmouseover="rs('polynom3/D10')" onmouseout="rh()">POLYNOM3:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c6">p</font> as    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c2">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E18">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">p</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E7:42"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T17" onmouseover="rs('uproots/T17')" onmouseout="rh()">UPROOTS:17</a></span>;<br/>
<a NAME="E8:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">lenq</font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  ;<br/>
<span class="lab"><font color="Green" title="E20">A5</font></span>: 1 = 
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T27" onmouseover="rs('hurwitz/T27')" onmouseout="rh()">HURWITZ:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#D2" onmouseover="rs('hurwitz/D2')" onmouseout="rh()">HURWITZ:def 2</a></span>
;<br/>
<a NAME="E11:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<div><span class="lab"><font color="Green" title="E22">A7</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2"><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#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> holds <br/><font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="c10">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> implies <font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">assume </span><a NAME="E1:42_2"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</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"> <font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">j</font> = <font color="Maroon" title="c10">i</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E3:42_2"/>
( <font color="Maroon" title="c11">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp; <font color="Maroon" title="c11">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E21">A6</font></span></a></span>;<br/><a NAME="E4:42_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T26" onmouseover="rs('hurwitz/T26')" onmouseout="rh()">HURWITZ:26</a></span>;<br/><span class="kw">hence </span><a NAME="E5:42_2"/>
<font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</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>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_3"><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">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a> <font color="Maroon" title="c8">p</font> holds <br/> <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="c10">m</font> <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a> <font color="Maroon" title="c8">p</font> implies  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:42_3"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a> <font color="Maroon" title="c8">p</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="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">j</font> = <font color="Maroon" title="c9">lenq</font> as    <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>  ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_3_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">  not  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">m</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_3_1"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">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"> contradiction</span><br/><a NAME="E2:42_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">lenq</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">m</font>
 ;<br/><a NAME="E3:42_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">lenq</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">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="E4:42_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p3">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E1:42_3"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="algseq_1.html#D2" onmouseover="rs('algseq_1/D2')" onmouseout="rh()">ALGSEQ_1:def 2</a></span>;<br/><span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1 = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">R</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T25" onmouseover="rs('hurwitz/T25')" onmouseout="rh()">HURWITZ:25</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:42_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_3_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</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><span class="kw">hence </span><a NAME="E4:42_3"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A11</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="algseq_1.html#D3" onmouseover="rs('algseq_1/D3')" onmouseout="rh()">ALGSEQ_1:def 3</a>, <a class="ref" href="algseq_1.html#D2" onmouseover="rs('algseq_1/D2')" onmouseout="rh()">ALGSEQ_1:def 2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_4"><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">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> holds <br/><font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> implies <font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:42_4"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</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"> <font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 = 
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#D2" onmouseover="rs('hurwitz/D2')" onmouseout="rh()">HURWITZ:def 2</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T27" onmouseover="rs('hurwitz/T27')" onmouseout="rh()">HURWITZ:27</a></span>
;<br/><a NAME="E3:42_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_4"><span class="lab"><font color="Green" title="E24">A12</font></span></a></span>;<br/><a NAME="E4:42_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_4_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_4_2"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  or <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_4_2_1"><span class="kw">suppose </span></a><a NAME="E1:42_4_2_1"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1: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"> <font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A15</font></span>: <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T25" onmouseover="rs('hurwitz/T25')" onmouseout="rh()">HURWITZ:25</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c3">a</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D7" onmouseover="rs('group_1/D7')" onmouseout="rh()">GROUP_1:def 7</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D7" onmouseover="rs('group_1/D7')" onmouseout="rh()">GROUP_1:def 7</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="field_4.html#E23"><span class="lab"><font color="Green" title="E15">Lm2</font></span></a></span>
;<br/><span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font> = 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_4_2_1"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="ref" href="hurwitz.html#T25" onmouseover="rs('hurwitz/T25')" onmouseout="rh()">HURWITZ:25</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<font color="Maroon" title="c4">b</font>,<a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D7" onmouseover="rs('group_1/D7')" onmouseout="rh()">GROUP_1:def 7</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D7" onmouseover="rs('group_1/D7')" onmouseout="rh()">GROUP_1:def 7</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">b</font>

;<br/><span class="kw">hence </span><a NAME="E4:42_4_2_1"/>
<font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_4_2_1"><span class="lab"><font color="Green" title="E27">A15</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_4_2_2"><span class="kw">suppose </span></a><a NAME="E1:42_4_2_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1: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"> <font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:42_4_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E3:42_4_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <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="E4:42_4_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A17</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_4"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1 = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">R</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#T25" onmouseover="rs('hurwitz/T25')" onmouseout="rh()">HURWITZ:25</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>
;<br/><span class="kw">hence </span><font color="Maroon" title="c8">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font> = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_4_2_2"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="#E4:42_4"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_4_2_2"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="hurwitz.html#T25" onmouseover="rs('hurwitz/T25')" onmouseout="rh()">HURWITZ:25</a></span>
;<br/> <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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:42"/>
 <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c3">a</font>) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c4">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="algseq_1.html#T12" onmouseover="rs('algseq_1/T12')" onmouseout="rh()">ALGSEQ_1:12</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>
