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<span class="kw">let </span><font color="Maroon" title="c1">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  implies <font color="Maroon" title="c1">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">p</font> = <font color="Maroon" title="c1">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:56"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">z</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c1">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> implies  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:56_1"/>
<font color="Maroon" title="c3">z</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c1">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E2:56_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A2</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">z</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="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></span>;<br/> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">z</font> = 
<font color="Maroon" title="c1">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>,<font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T31" title="POLYNOM5:th.31">POLYNOM5:31</a></span>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</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="polynom4.html#T21" title="POLYNOM4:th.21">POLYNOM4:21</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:56_1"/>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:56"/>
<font color="Maroon" title="c1">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> is  <a href="hurwitz.html#V1" title="HURWITZ:attr.1">Hurwitz</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#D8" target="_self" title="HURWITZ:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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