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

<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   non  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</a>   <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">F</font></span>)</span>  ex <font color="Olive" title="b2">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font> st <font color="Olive" title="b1">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b2">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   non  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</a>   <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">F</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">  ex <font color="Olive" title="b1">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font> st <font color="Maroon" title="c2">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b1">E</font></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">F</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><br/>  for <font color="Olive" title="b2">f</font> being   non  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</a>   <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="Olive" title="b1">F</font></span>)</span>  st  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b3">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Olive" title="b1">F</font> st <font color="Olive" title="b2">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b3">E</font>;<br/>
<a NAME="E1:72"/><span class="lab"><font color="Green" title="E30">IA</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   non  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</a>   <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="c3">F</font></span>)</span>  st  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> ex <font color="Olive" title="b2">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c3">F</font> st <font color="Olive" title="b1">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b2">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   non  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</a>   <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="c3">F</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"> (  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  ex <font color="Olive" title="b1">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c3">F</font> st <font color="Maroon" title="c4">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b1">E</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:72_1"/><span class="lab"><font color="Green" title="E31">A</font></span>: 
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  ex <font color="Olive" title="b1">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c3">F</font> st <font color="Maroon" title="c4">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b1">E</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">E</font> = <font color="Maroon" title="c3">F</font> as   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c3">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T6" onmouseover="rs('field_4/T6')" onmouseout="rh()">FIELD_4:6</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">E</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="c4">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Maroon" title="c5">E</font></span><br/>

<span class="kw">thus </span><a NAME="E3:72_1"/>
<font color="Maroon" title="c4">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Maroon" title="c5">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T29" onmouseover="rs('field_4/T29')" onmouseout="rh()">FIELD_4:29</a>, <a class="txt" href="#E1:72_1"><span class="lab"><font color="Green" title="E31">A</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="lab"><font color="Green" title="E31">IS</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_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">k</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">k</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <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">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:72_2"/><span class="lab"><font color="Green" title="E32">IV</font></span>: 
<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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_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">  for <font color="Olive" title="b1">F</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><br/>  for <font color="Olive" title="b2">f</font> being   non  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</a>   <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="Olive" title="b1">F</font></span>)</span>  st  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b3">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Olive" title="b1">F</font> st <font color="Olive" title="b2">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b3">E</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   non  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</a>   <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="c4">F</font></span>)</span>  st  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 holds <br/> ex <font color="Olive" title="b2">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Olive" title="b2">E</font> st <font color="Olive">b<sub>3</sub></font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive">b<sub>4</sub></font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   non  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</a>   <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="c4">F</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"> (  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies  ex <font color="Olive" title="b1">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Olive" title="b1">E</font> st <font color="Olive">b<sub>2</sub></font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive">b<sub>3</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:72_2_1"/><span class="lab"><font color="Green" title="E33">AS2</font></span>: 
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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">  ex <font color="Olive" title="b1">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Olive" title="b1">E</font> st <font color="Olive">b<sub>2</sub></font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive">b<sub>3</sub></font></span><br/><span class="kw">consider </span><font color="Maroon" title="c6">E1</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c4">F</font><span class="kw"> such that </span><br/><a NAME="E3:72_2_1"/><span class="lab"><font color="Green" title="E34">A</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="field_4.html#R2" title="FIELD_4:pred.2">is_with_roots_in</a> <font color="Maroon" title="c6">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#T30" target="_self" onmouseover="rs('field_5/T30')" onmouseout="rh()">main</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c5">f</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c6">E1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T8" onmouseover="rs('field_4/T8')" onmouseout="rh()">FIELD_4:8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c7">f1</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="c6">E1</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">consider </span><font color="Maroon" title="c9">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">E1</font><span class="kw"> such that </span><br/><a NAME="E7:72_2_1"/><span class="lab"><font color="Green" title="E35">B</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="field_4.html#R1" title="FIELD_4:pred.1">is_a_root_of</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_2_1"><span class="lab"><font color="Green" title="E34">A</font></span></a>, <a class="ref" href="field_4.html#D3" onmouseover="rs('field_4/D3')" onmouseout="rh()">FIELD_4:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">h</font> =  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c9">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="c6">E1</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="c11">q</font> =  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c9">a</font>) as    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of <font color="Maroon" title="c6">E1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_5.html#T51" onmouseover="rs('ring_5/T51')" onmouseout="rh()">RING_5:51</a></span>;<br/><a NAME="E10:72_2_1"/>
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c9">a</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="c6">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:72_2_1"><span class="lab"><font color="Green" title="E35">B</font></span></a>, <a class="ref" href="field_4.html#D2" onmouseover="rs('field_4/D2')" onmouseout="rh()">FIELD_4:def 2</a></span>;<br/><a NAME="E11:72_2_1"/><span class="kw">then </span>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c8">f1</font>,<font color="Maroon" title="c9">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T26" onmouseover="rs('field_4/T26')" onmouseout="rh()">FIELD_4:26</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">g2</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c6">E1</font><span class="kw"> such that </span><br/><a NAME="E13:72_2_1"/><span class="lab"><font color="Green" title="E36">C1</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">h</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c12">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#T1" onmouseover="rs('ring_4/T1')" onmouseout="rh()">RING_4:1</a>, <a class="ref" href="ring_5.html#T11" onmouseover="rs('ring_5/T11')" onmouseout="rh()">RING_5:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c12">g2</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="c6">E1</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/><a NAME="E15:72_2_1"/><span class="lab"><font color="Green" title="E37">C</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:72_2_1"><span class="lab"><font color="Green" title="E36">C1</font></span></a>, <a class="ref" href="polynom3.html#D10" onmouseover="rs('polynom3/D10')" onmouseout="rh()">POLYNOM3:def 10</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_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="E1:72_2_1_1"/>
( <font color="Maroon" title="c13">g</font> is  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</a>  or  not <font color="Maroon" title="c13">g</font> is  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:72_2_1_1_1"/>
<font color="Maroon" title="c13">g</font> is  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</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">  ex <font color="Olive" title="b1">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Olive" title="b1">E</font> st <font color="Olive">b<sub>2</sub></font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive">b<sub>3</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">E1</font><span class="kw"> such that </span><br/><a NAME="E3:72_2_1_1_1"/><span class="lab"><font color="Green" title="E38">D</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">e</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c6">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#T20" onmouseover="rs('ring_4/T20')" onmouseout="rh()">RING_4:20</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_2_1_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">  not <font color="Maroon" title="c14">e</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:72_2_1_1_1_1"/>
<font color="Maroon" title="c14">e</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</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"> contradiction</span><br/><a NAME="E2:72_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c6">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_2_1_1_1"><span class="lab"><font color="Green" title="E38">D</font></span></a>, <a class="ref" href="ring_4.html#T13" onmouseover="rs('ring_4/T13')" onmouseout="rh()">RING_4:13</a></span>;<br/><a NAME="E3:72_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:72_2_1"><span class="lab"><font color="Green" title="E36">C1</font></span></a>, <a class="ref" href="field_4.html#T12" onmouseover="rs('field_4/T12')" onmouseout="rh()">FIELD_4:12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:72_2_1_1_1_1"/>
contradiction
 ; <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">e</font> = <font color="Maroon" title="c14">e</font> as   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">E1</font> ;<br/><font color="Maroon" title="c5">f</font> = 
<span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">e</font> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">1_.</a> <font color="Maroon" title="c6">E1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_2_1_1_1"><span class="lab"><font color="Green" title="E38">D</font></span></a>, <a class="txt" href="#E13:72_2_1"><span class="lab"><font color="Green" title="E36">C1</font></span></a>, <a class="ref" href="ring_4.html#T16" onmouseover="rs('ring_4/T16')" onmouseout="rh()">RING_4:16</a></span>
<br/>.= 
<font color="Maroon" title="c15">e</font> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c9">a</font>)</span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p2">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">1_.</a> <font color="Maroon" title="c6">E1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#T10" onmouseover="rs('ring_4/T10')" onmouseout="rh()">RING_4:10</a></span>
<br/>.= 
<font color="Maroon" title="c15">e</font> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <font color="Maroon" title="c11">q</font>

;<br/><span class="kw">hence </span><a NAME="E7:72_2_1_1_1"/>
 ex <font color="Olive" title="b1">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c4">F</font> st <font color="Maroon" title="c5">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D5" onmouseover="rs('field_4/D5')" onmouseout="rh()">FIELD_4:def 5</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="72_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:72_2_1_1_2"/><span class="lab"><font color="Green" title="E38">I</font></span>: 
 not <font color="Maroon" title="c13">g</font> is  <a href="ring_4.html#V3" title="RING_4:attr.3">constant</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">  ex <font color="Olive" title="b1">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Olive" title="b1">E</font> st <font color="Olive">b<sub>2</sub></font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive">b<sub>3</sub></font></span><br/><div class="add"><a NAME="E2:72_2_1_1_2"/>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="72_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c14">g1</font> = <font color="Maroon" title="c13">g</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c6">E1</font> ;<br/>
<a NAME="E2:72_2_1_1_2_1"/>
<font color="Maroon" title="c14">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c6">E1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_2_1_1_2"><span class="lab"><font color="Green" title="E38">I</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E39">D</font></span>:  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c8">f1</font> = 
<span class="p1">(<span class="default"><a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <span class="p2">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c9">a</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c14">g1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:72_2_1"><span class="lab"><font color="Green" title="E36">C1</font></span></a>, <a class="ref" href="hurwitz.html#T23" onmouseover="rs('hurwitz/T23')" onmouseout="rh()">HURWITZ:23</a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c14">g1</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/>
<a NAME="E4:72_2_1_1_2_1"/>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_2_1"><span class="lab"><font color="Green" title="E33">AS2</font></span></a>, <a class="ref" href="field_4.html#T20" onmouseover="rs('field_4/T20')" onmouseout="rh()">FIELD_4:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:72_2_1_1_2_1"/>
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_2_1_1_2_1"><span class="lab"><font color="Green" title="E39">D</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><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">E2</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c6">E1</font><span class="kw"> such that </span><br/><a NAME="E4:72_2_1_1_2"/><span class="lab"><font color="Green" title="E39">E</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Maroon" title="c14">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_2_1_1_2"><span class="lab"><font color="Green" title="E38">I</font></span></a>, <a class="txt" href="#E1:72_2"><span class="lab"><font color="Green" title="E32">IV</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">E2</font> = <font color="Maroon" title="c14">E2</font> as   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c4">F</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c16">e</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">E2</font>, <font color="Maroon" title="c17">r</font> being    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of <font color="Maroon" title="c15">E2</font><span class="kw"> such that </span><br/><a NAME="E7:72_2_1_1_2"/><span class="lab"><font color="Green" title="E40">F</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c16">e</font> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <font color="Maroon" title="c17">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_2_1_1_2"><span class="lab"><font color="Green" title="E39">E</font></span></a>, <a class="ref" href="field_4.html#D5" onmouseover="rs('field_4/D5')" onmouseout="rh()">FIELD_4:def 5</a></span>;<br/><a NAME="E8:72_2_1_1_2"/>
<font color="Maroon" title="c6">E1</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c15">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D1" onmouseover="rs('field_4/D1')" onmouseout="rh()">FIELD_4:def 1</a></span>;<br/><a NAME="E9:72_2_1_1_2"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">E1</font> <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 <font color="Maroon" title="c15">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#D3" onmouseover="rs('c0sp1/D3')" onmouseout="rh()">C0SP1:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">a1</font> = <font color="Maroon" title="c9">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">E2</font> ;<br/><a NAME="E11:72_2_1_1_2"/>
 <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c18">a1</font>) is    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of <font color="Maroon" title="c15">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_5.html#T51" onmouseover="rs('ring_5/T51')" onmouseout="rh()">RING_5:51</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">u</font> = <span class="p1">(<span class="default"><a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c18">a1</font>)</span>)</span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c17">r</font> as    <a href="ring_5.html#M1" title="RING_5:mode.1">Ppoly</a> of <font color="Maroon" title="c15">E2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_5.html#T52" onmouseover="rs('ring_5/T52')" onmouseout="rh()">RING_5:52</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">q</font> =  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c9">a</font>) as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c15">E2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T8" onmouseover="rs('field_4/T8')" onmouseout="rh()">FIELD_4:8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">q1</font> = <font color="Maroon" title="c20">q</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="c15">E2</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="c22">eg</font> = <font color="Maroon" title="c16">e</font> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <font color="Maroon" title="c17">r</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="c15">E2</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/><a NAME="E16:72_2_1_1_2"/><span class="lab"><font color="Green" title="E41">Y</font></span>: 
<font color="Maroon" title="c20">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="hurwitz.html#K3" title="HURWITZ:func.3">rpoly</a> (1,<font color="Maroon" title="c18">a1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T21" onmouseover="rs('field_4/T21')" onmouseout="rh()">FIELD_4:21</a></span>;<br/><a NAME="E17:72_2_1_1_2"/><span class="lab"><font color="Green" title="E42">Z1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c13">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c6">E1</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c6">E1</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/><a NAME="E18:72_2_1_1_2"/>
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c15">E2</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</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="c6">E1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c6">E1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T18" onmouseover="rs('field_4/T18')" onmouseout="rh()">FIELD_4:18</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">f</font> = 
<font color="Maroon" title="c21">q1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c22">eg</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:72_2_1_1_2"><span class="lab"><font color="Green" title="E40">F</font></span></a>, <a class="txt" href="#E15:72_2_1"><span class="lab"><font color="Green" title="E37">C</font></span></a>, <a class="txt" href="#E17:72_2_1_1_2"><span class="lab"><font color="Green" title="E42">Z1</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c20">q</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">e</font> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <font color="Maroon" title="c17">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/>.= 
<font color="Maroon" title="c16">e</font> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <font color="Maroon" title="c19">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:72_2_1_1_2"><span class="lab"><font color="Green" title="E41">Y</font></span></a>, <a class="ref" href="ring_4.html#T10" onmouseover="rs('ring_4/T10')" onmouseout="rh()">RING_4:10</a></span>
;<br/><span class="kw">hence </span><a NAME="E20:72_2_1_1_2"/>
 ex <font color="Olive" title="b1">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c4">F</font> st <font color="Maroon" title="c5">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D5" onmouseover="rs('field_4/D5')" onmouseout="rh()">FIELD_4:def 5</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="E3:72_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:72"/><span class="lab"><font color="Green" title="E32">I</font></span>: 
 for <font color="Olive" title="b1">k</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <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#S10" onmouseover="rs('nat_1/S10')" onmouseout="rh()">NAT_1:sch 10</a>(<a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E30">IA</font></span></a>, <a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E31">IS</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="E5:72"/><span class="lab"><font color="Green" title="E33">H</font></span>: 
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 ;<br/>
<a NAME="E6:72"/>
<font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72"><span class="lab"><font color="Green" title="E33">H</font></span></a>, <a class="ref" href="ring_4.html#D4" onmouseover="rs('ring_4/D4')" onmouseout="rh()">RING_4:def 4</a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:72"/>
 ex <font color="Olive" title="b1">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font> st <font color="Maroon" title="c2">f</font> <a href="field_4.html#R3" title="FIELD_4:pred.3">splits_in</a> <font color="Olive" title="b1">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E32">I</font></span></a>, <a class="txt" href="#E5:72"><span class="lab"><font color="Green" title="E33">H</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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