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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="hurwitz2.html#V6" title="HURWITZ2:attr.6">real</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</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 <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </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 <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</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"> (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st <font color="Olive" title="b1">p</font> is  <a href="hurwitz2.html#V6" title="HURWITZ2:attr.6">real</a>  &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Olive" title="b1">p</font></span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> ;<br/>
<div><span class="lab"><font color="Green" title="E31">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_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">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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"> ( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:62_1"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1_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">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st <font color="Olive" title="b1">p</font> is  <a href="hurwitz2.html#V6" title="HURWITZ2:attr.6">real</a>  &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> holds <br/> for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Olive" title="b1">p</font></span>)</span>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</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="c4">p</font> is  <a href="hurwitz2.html#V6" title="HURWITZ2:attr.6">real</a>  &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> implies  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:62_1_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
( <font color="Maroon" title="c4">p</font> is  <a href="hurwitz2.html#V6" title="HURWITZ2:attr.6">real</a>  &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1_1_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"> ( ( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <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="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  ) ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1"><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="E1:62_1_1_1_1"/>
( <font color="Maroon" title="c3">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="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:62_1_1_1_1_1"/>
<font color="Maroon" title="c3">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">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span><br/><div class="add"><a NAME="E2:62_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="polynom4.html#T5" onmouseover="rs('polynom4/T5')" onmouseout="rh()">POLYNOM4:5</a></span>;<br/><a NAME="E3:62_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz2.html#T7" target="_self" onmouseover="rs('hurwitz2/T7')" onmouseout="rh()">Th7</a></span>;<br/><span class="kw">thus </span><a NAME="E4:62_1_1_1_1_1"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</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"> (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:62_1_1_1_1_1_1"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c5">x</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">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span><br/>

 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c5">x</font>) = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="polynom4.html#T17" onmouseover="rs('polynom4/T17')" onmouseout="rh()">POLYNOM4:17</a></span>
<br/>.= 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:62_1_1_1_1_1_1"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T4" onmouseover="rs('complex1/T4')" onmouseout="rh()">COMPLEX1:4</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:62_1_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <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 <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">LMp</font> =  <a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font>;<br/><span class="p1">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <font color="Maroon" title="c4">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T26" onmouseover="rs('polynom3/T26')" onmouseout="rh()">POLYNOM3:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <font color="Maroon" title="c4">p</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <font color="Maroon" title="c4">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T29" onmouseover="rs('polynom3/T29')" onmouseout="rh()">POLYNOM3:29</a></span>
<br/>.= 
<font color="Maroon" title="c4">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T28" onmouseover="rs('polynom3/T28')" onmouseout="rh()">POLYNOM3:28</a></span>
;<br/><a NAME="E3:62_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A6</font></span>: 
 <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz2.html#T16" target="_self" onmouseover="rs('hurwitz2/T16')" onmouseout="rh()">Th16</a></span>;<br/><span class="kw">thus </span><a NAME="E4:62_1_1_1_1_2"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</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"> (  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:62_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c6">x</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">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">t</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a><span class="kw"> such that </span><br/><a NAME="E3:62_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
(  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c7">t</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> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">t</font> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span> &amp; (  for <font color="Olive" title="b1">n</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">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/><font color="Maroon" title="c7">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="polynom4.html#T16" onmouseover="rs('polynom4/T16')" onmouseout="rh()">POLYNOM4:16</a></span>;<br/>
<span class="lab"><font color="Green" title="E38">A9</font></span>: <font color="Maroon" title="c4">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p1">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span> = 
<font color="Maroon" title="c7">t</font> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="polynom3.html#T26" onmouseover="rs('polynom3/T26')" onmouseout="rh()">POLYNOM3:26</a></span>
<br/>.= 
<font color="Maroon" title="c7">t</font> <a href="polynom3.html#K7" title="POLYNOM3:func.7">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T29" onmouseover="rs('polynom3/T29')" onmouseout="rh()">POLYNOM3:29</a></span>
<br/>.= 
<font color="Maroon" title="c7">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T28" onmouseover="rs('polynom3/T28')" onmouseout="rh()">POLYNOM3:28</a></span>
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1_2_2_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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c7">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">n</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">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a></span><br/><a NAME="E1:62_1_1_1_1_2_2_2"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<font color="Maroon" title="c8">n</font> <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="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1_2_2_2_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"> ( ( <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c7">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ) or ( <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c7">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1_2_2_2_1_1"><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="E1:62_1_1_1_1_2_2_2_1_1"/>
( <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1_2_2_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:62_1_1_1_1_2_2_2_1_1_1"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <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">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a></span><br/><div class="add"><a NAME="E2:62_1_1_1_1_2_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E1:62_1_1_1_1_2_2_2"><span class="lab"><font color="Green" title="E39">A10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:62_1_1_1_1_2_2_2_1_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E33">A3</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="62_1_1_1_1_2_2_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:62_1_1_1_1_2_2_2_1_1_2"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <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">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">lp</font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><a NAME="E3:62_1_1_1_1_2_2_2_1_1_2"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c7">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">lp</font> <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="#E3:62_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a></span>;<br/><a NAME="E4:62_1_1_1_1_2_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">lp</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> <font color="Maroon" title="c7">t</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="E5:62_1_1_1_1_2_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_1_1_2_2_2_1_1_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="ref" href="algseq_1.html#T8" onmouseover="rs('algseq_1/T8')" onmouseout="rh()">ALGSEQ_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E6:62_1_1_1_1_2_2_2_1_1_2"/>
<font color="Maroon" title="c7">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:62_1_1_1_1_2_2_2"/>
<font color="Maroon" title="c7">t</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:62_1_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon" title="c7">t</font> is  <a href="hurwitz2.html#V6" title="HURWITZ2:attr.6">real</a> 
 
;<br/>
<a NAME="E7:62_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E40">A13</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1_2_2_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1_2_2_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"> ( (  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">p</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  ) or (  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">p</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1_2_2_3_1_1"><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="E1:62_1_1_1_1_2_2_3_1_1"/>
(  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">p</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  or  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">p</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1_1_2_2_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:62_1_1_1_1_2_2_3_1_1_1"/>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">p</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</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">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span><br/><div class="add"><a NAME="E2:62_1_1_1_1_2_2_3_1_1_1"/><span class="kw">then </span>
 <a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p1">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz2.html#T19" target="_self" onmouseover="rs('hurwitz2/T19')" onmouseout="rh()">Th19</a></span>;<br/><span class="kw">then </span> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>) = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T17" onmouseover="rs('polynom4/T17')" onmouseout="rh()">POLYNOM4:17</a></span>
<br/>.= 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" onmouseover="rs('complfld/D1')" onmouseout="rh()">COMPLFLD:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:62_1_1_1_1_2_2_3_1_1_1"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T4" onmouseover="rs('complex1/T4')" onmouseout="rh()">COMPLEX1:4</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="62_1_1_1_1_2_2_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:62_1_1_1_1_2_2_3_1_1_2"/><span class="lab"><font color="Green" title="E41">A14</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">p</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> </span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A15</font></span>:  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>) = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c6">x</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz2.html#T20" target="_self" onmouseover="rs('hurwitz2/T20')" onmouseout="rh()">Th20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c6">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T22" onmouseover="rs('polynom4/T22')" onmouseout="rh()">POLYNOM4:22</a></span>
;<br/><span class="kw">set </span><font color="Maroon" title="c8">z1</font> = <font color="Maroon" title="c4">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> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">z2</font> = <span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c6">x</font>,<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> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>);<br/><a NAME="E3:62_1_1_1_1_2_2_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></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>  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E1:62_1_1_1_1_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>;<br/><a NAME="E4:62_1_1_1_1_2_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A16</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</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> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c6">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E1:62_1_1_1_1_2_2_3_1_1_2"><span class="lab"><font color="Green" title="E41">A14</font></span></a>, <a class="ref" href="hurwitz2.html#T6" target="_self" onmouseover="rs('hurwitz2/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E5:62_1_1_1_1_2_2_3_1_1_2"/>
<font color="Maroon" title="c4">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> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/><a NAME="E6:62_1_1_1_1_2_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A17</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D2" onmouseover="rs('complex1/D2')" onmouseout="rh()">COMPLEX1:def 2</a></span>;<br/><span class="kw">thus </span> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c6">x</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c6">x</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62_1_1_1_1_2_2_3_1_1_2"><span class="lab"><font color="Green" title="E42">A15</font></span></a>, <a class="ref" href="complex1.html#T9" onmouseover="rs('complex1/T9')" onmouseout="rh()">COMPLEX1:9</a></span>
<br/>.= 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_1_1_1_1_2_2_3_1_1_2"><span class="lab"><font color="Green" title="E44">A17</font></span></a>, <a class="txt" href="#E4:62_1_1_1_1_2_2_3_1_1_2"><span class="lab"><font color="Green" title="E43">A16</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:62_1_1_1_1_2_2_3"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> = 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p4">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="polynom4.html#T19" onmouseover="rs('polynom4/T19')" onmouseout="rh()">POLYNOM4:19</a></span>
<br/>.= 
<a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p3">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">-</a> <span class="p5">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:62_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E40">A13</font></span></a>, <a class="ref" href="complex1.html#T8" onmouseover="rs('complex1/T8')" onmouseout="rh()">COMPLEX1:8</a></span>
<br/>.= 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E3:62_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E4:62_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E1:62_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E1:62_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E1:62_1"><span class="lab"><font color="Green" title="E32">A2</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><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="E3:62_1_1"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  st  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  holds <br/> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <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"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:62_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">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 NAME="E2:62"/><span class="lab"><font color="Green" title="E32">A18</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" onmouseover="rs('nat_1/S4')" onmouseout="rh()">NAT_1:sch 4</a>(<a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E31">A1</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:62"/><span class="lab"><font color="Green" title="E33">A19</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 ;<br/>
<span class="kw">thus </span><a NAME="E5:62"/>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p2">(<span class="default"><a href="hurwitz2.html#K2" title="HURWITZ2:func.2">odd_part</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E33">A19</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/>


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