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<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">p</font> being   <a href="ring_2.html#V5" title="RING_2:attr.5">irreducible</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">p</font> <a href="field_4.html#R2" title="FIELD_4:pred.2">is_with_roots_in</a> <font color="Olive" title="b2">E</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="ring_2.html#V5" title="RING_2:attr.5">irreducible</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">p</font> <a href="field_4.html#R2" title="FIELD_4:pred.2">is_with_roots_in</a> <font color="Olive" title="b1">E</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">KF</font> =  <a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">p</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">u</font> =  <a href="field_1.html#K7" title="FIELD_1:func.7">KrRoot</a> <font color="Maroon" title="c2">p</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">FP</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><span class="kw"> such that </span><br/><a NAME="E2:70"/><span class="lab"><font color="Green" title="E29">X1</font></span>: 
(  <a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">p</font>),<font color="Maroon" title="c5">FP</font> <a href="ring_3.html#R2" title="RING_3:pred.2">are_isomorphic</a>  &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">FP</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<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="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#T29" target="_self" onmouseover="rs('field_5/T29')" onmouseout="rh()">thisofield</a></span>;<br/>
<a NAME="E3:70"/><span class="lab"><font color="Green" title="E30">X</font></span>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">FP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">FP</font> )
 
;<br/>
<a NAME="E4:70"/><span class="lab"><font color="Green" title="E31">X2</font></span>: 
<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c5">FP</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">are_disjoint</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1"><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="70_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  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">FP</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:70_1_1"/><span class="lab"><font color="Green" title="E32">A</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">FP</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</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/><span class="kw">set </span><font color="Maroon" title="c6">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">FP</font>;<br/><a NAME="E2:70_1_1"/><span class="lab"><font color="Green" title="E33">B</font></span>: 
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">FP</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">FP</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">FP</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E32">A</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:70_1_1"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">FP</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:70_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E33">B</font></span></a>, <a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E29">X1</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 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>
<span class="kw">hence </span><a NAME="E2:70_1"/>
<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c5">FP</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">are_disjoint</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70"><span class="lab"><font color="Green" title="E30">X</font></span></a>, <a class="ref" href="field_2.html#D1" onmouseover="rs('field_2/D1')" onmouseout="rh()">FIELD_2: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>
<span class="kw">consider </span><font color="Maroon" title="c6">phi</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c5">FP</font><span class="kw"> such that </span><br/><a NAME="E6:70"/><span class="lab"><font color="Green" title="E32">X3</font></span>: 
<font color="Maroon" title="c6">phi</font> is  <a href="mod_4.html#V7" title="MOD_4:attr.7">isomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70"><span class="lab"><font color="Green" title="E29">X1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">KroneckerIsop</font> = <span class="p1">(<span class="default"><a href="field_4.html#K4" title="FIELD_4:func.4">KroneckerIso</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="field_2.html#K1" title="FIELD_2:func.1">"</a>  as   <a href="ring_3.html#NM3" title="RING_3:NM.3">Isomorphism</a> of <span class="p1">(<span class="default"><a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_3.html#T73" onmouseover="rs('ring_3/T73')" onmouseout="rh()">RING_3:73</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c6">phi</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c5">FP</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T13" onmouseover="rs('funct_2/T13')" onmouseout="rh()">FUNCT_2:13</a></span>;<br/>
<a NAME="E9:70"/><span class="lab"><font color="Green" title="E33">X4</font></span>: 
( <font color="Maroon" title="c9">h</font> is  <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>  &amp; <font color="Maroon" title="c9">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E32">X3</font></span></a>, <a class="ref" href="ringcat1.html#T1" onmouseover="rs('ringcat1/T1')" onmouseout="rh()">RINGCAT1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">FP</font> = <font color="Maroon" title="c5">FP</font> as  <font color="Maroon" title="c1">F</font> <a href="ring_3.html#V1" title="RING_3:attr.1">-monomorphic</a>  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_3.html#D3" onmouseover="rs('ring_3/D3')" onmouseout="rh()">RING_3:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c9">h</font> as   <a href="ring_3.html#NM1" title="RING_3:NM.1">Monomorphism</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c10">FP</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70"><span class="lab"><font color="Green" title="E33">X4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">E</font> =  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <font color="Maroon" title="c11">h</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E31">X2</font></span></a>, <a class="ref" href="field_2.html#T12" onmouseover="rs('field_2/T12')" onmouseout="rh()">FIELD_2:12</a></span>;<br/>
<a NAME="E13:70"/>
 <a href="field_2.html#K8" title="FIELD_2:func.8">emb_iso</a> <font color="Maroon" title="c11">h</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E31">X2</font></span></a>, <a class="ref" href="field_2.html#T15" onmouseover="rs('field_2/T15')" onmouseout="rh()">FIELD_2:15</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">embisoh</font> = <span class="p1">(<span class="default"><a href="field_2.html#K8" title="FIELD_2:func.8">emb_iso</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c10">FP</font>,<font color="Maroon" title="c12">E</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T25" onmouseover="rs('funct_2/T25')" onmouseout="rh()">FUNCT_2:25</a></span>;<br/>
<a NAME="E15:70"/><span class="lab"><font color="Green" title="E34">Y</font></span>: 
(  <a href="field_2.html#K8" title="FIELD_2:func.8">emb_iso</a> <font color="Maroon" title="c11">h</font> is  <a href="ringcat1.html#V1" title="RINGCAT1:attr.1">linear</a>  &amp;  <a href="field_2.html#K8" title="FIELD_2:func.8">emb_iso</a> <font color="Maroon" title="c11">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="field_2.html#K8" title="FIELD_2:func.8">emb_iso</a> <font color="Maroon" title="c11">h</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70"><span class="lab"><font color="Green" title="E31">X2</font></span></a>, <a class="ref" href="field_2.html#T13" onmouseover="rs('field_2/T13')" onmouseout="rh()">FIELD_2:13</a>, <a class="ref" href="field_2.html#T14" onmouseover="rs('field_2/T14')" onmouseout="rh()">FIELD_2:14</a>, <a class="ref" href="field_2.html#T15" onmouseover="rs('field_2/T15')" onmouseout="rh()">FIELD_2:15</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">FP</font> = <font color="Maroon" title="c10">FP</font> as  <font color="Maroon" title="c12">E</font> <a href="ring_3.html#V2" title="RING_3:attr.2">-isomorphic</a>  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_3.html#D4" onmouseover="rs('ring_3/D4')" onmouseout="rh()">RING_3:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">embisoh</font> = <font color="Maroon" title="c13">embisoh</font> as   <a href="ring_3.html#NM3" title="RING_3:NM.3">Isomorphism</a> of <font color="Maroon" title="c14">FP</font>,<font color="Maroon" title="c12">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:70"><span class="lab"><font color="Green" title="E34">Y</font></span></a>, <a class="ref" href="ring_3.html#T73" onmouseover="rs('ring_3/T73')" onmouseout="rh()">RING_3:73</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">iso</font> = <font color="Maroon" title="c15">embisoh</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">phi</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">iso</font> = <font color="Maroon" title="c15">embisoh</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">phi</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c12">E</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T13" onmouseover="rs('funct_2/T13')" onmouseout="rh()">FUNCT_2:13</a></span>;<br/>
<a NAME="E19:70"/><span class="lab"><font color="Green" title="E35">X5</font></span>: 
<font color="Maroon" title="c17">iso</font> is  <a href="quofield.html#NV1" title="QUOFIELD:NV.1">RingHomomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70"><span class="lab"><font color="Green" title="E32">X3</font></span></a>, <a class="ref" href="ringcat1.html#T1" onmouseover="rs('ringcat1/T1')" onmouseout="rh()">RINGCAT1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">E</font> = <font color="Maroon" title="c12">E</font> as   <a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">p</font>) <a href="ring_2.html#V3" title="RING_2:attr.3">-homomorphic</a>  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_2.html#D4" onmouseover="rs('ring_2/D4')" onmouseout="rh()">RING_2:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">iso</font> = <font color="Maroon" title="c17">iso</font> as   <a href="ring_2.html#NM2" title="RING_2:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c18">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70"><span class="lab"><font color="Green" title="E35">X5</font></span></a></span>;<br/>
<a NAME="E22:70"/>
 <a href="field_1.html#K7" title="FIELD_1:func.7">KrRoot</a> <font color="Maroon" title="c2">p</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a>  <a href="field_1.html#K6" title="FIELD_1:func.6">emb</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_1.html#T42" onmouseover="rs('field_1/T42')" onmouseout="rh()">FIELD_1:42</a></span>;<br/>
<a NAME="E23:70"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">Z</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K3" title="FIELD_1:func.3">PolyHom</a> <font color="Maroon" title="c19">iso</font></span>)</span> <a href="field_1.html#K2" title="FIELD_1:func.2">.</a> <span class="p2">(<span class="default"><a href="field_1.html#K6" title="FIELD_1:func.6">emb</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">iso</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="field_1.html#K7" title="FIELD_1:func.7">KrRoot</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c18">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_1.html#T33" onmouseover="rs('field_1/T33')" onmouseout="rh()">FIELD_1:33</a>, <a class="ref" href="polynom5.html#D7" onmouseover="rs('polynom5/D7')" onmouseout="rh()">POLYNOM5:def 7</a></span>;<br/>
<a NAME="E24:70"/>
<font color="Maroon" title="c1">F</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c18">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_2.html#T17" onmouseover="rs('field_2/T17')" onmouseout="rh()">FIELD_2:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">E</font> = <font color="Maroon" title="c18">E</font> as   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#T7" onmouseover="rs('field_4/T7')" onmouseout="rh()">FIELD_4:7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c20">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="c2">p</font> <a href="field_4.html#R2" title="FIELD_4:pred.2">is_with_roots_in</a> <font color="Maroon" title="c20">E</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c21">a</font> = <font color="Maroon" title="c19">iso</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="field_1.html#K7" title="FIELD_1:func.7">KrRoot</a> <font color="Maroon" title="c2">p</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c20">E</font> ;<br/>
<a NAME="E27:70"/>
<span class="p1">(<span class="default"><a href="field_1.html#K3" title="FIELD_1:func.3">PolyHom</a> <font color="Maroon" title="c19">iso</font></span>)</span> <a href="field_1.html#K2" title="FIELD_1:func.2">.</a> <span class="p1">(<span class="default"><a href="field_1.html#K6" title="FIELD_1:func.6">emb</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c22">g</font> = <span class="p1">(<span class="default"><a href="field_1.html#K3" title="FIELD_1:func.3">PolyHom</a> <font color="Maroon" title="c19">iso</font></span>)</span> <a href="field_1.html#K2" title="FIELD_1:func.2">.</a> <span class="p1">(<span class="default"><a href="field_1.html#K6" title="FIELD_1:func.6">emb</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>;<br/>
<a NAME="E1:70_2"/><span class="lab"><font color="Green" title="E37">A</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> holds  <font color="Maroon" title="c19">iso</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</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="c19">iso</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">a</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c24">b</font> = <font color="Maroon" title="c23">a</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font> as    <a href="ring_4.html#M1" title="RING_4:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</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="c25">v</font> =  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> (<span class="p2">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c24">b</font>) as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_1.html#T12" onmouseover="rs('ring_1/T12')" onmouseout="rh()">RING_1:12</a></span>;<br/>
<a NAME="E3:70_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E38">C</font></span>: <font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">a</font> = 
<font color="Maroon" title="c6">phi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c6">phi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_1.html#T39" onmouseover="rs('field_1/T39')" onmouseout="rh()">FIELD_1:39</a></span>
;<br/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <font color="Maroon" title="c11">h</font></span>)</span> = 
 <a href="field_2.html#K2" title="FIELD_2:func.2">carr</a> <font color="Maroon" title="c11">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_2.html#D7" onmouseover="rs('field_2/D7')" onmouseout="rh()">FIELD_2:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c14">FP</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">h</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_2.html#D2" onmouseover="rs('field_2/D2')" onmouseout="rh()">FIELD_2:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">FP</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">h</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>

;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">a1</font> = <font color="Maroon" title="c23">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <font color="Maroon" title="c11">h</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E7:70_2_1"/>
<font color="Maroon" title="c23">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c1">F</font>
 
;<br/>
<a NAME="E8:70_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">D</font></span>: 
<span class="p1">(<span class="default"><a href="field_2.html#K8" title="FIELD_2:func.8">emb_iso</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">phi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_2_1"><span class="lab"><font color="Green" title="E38">C</font></span></a>, <a class="ref" href="field_2.html#D8" onmouseover="rs('field_2/D8')" onmouseout="rh()">FIELD_2:def 8</a></span>;<br/>
<a NAME="E9:70_2_1"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="field_1.html#K4" title="FIELD_1:func.4">KroneckerField</a> (<font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:70_2_1"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="field_2.html#K8" title="FIELD_2:func.8">emb_iso</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c20">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c19">iso</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">a</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">embisoh</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_1.html#T39" onmouseover="rs('field_1/T39')" onmouseout="rh()">FIELD_1:39</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_2.html#K8" title="FIELD_2:func.8">emb_iso</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">phi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:70_2_1"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c23">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:70_2_1"><span class="lab"><font color="Green" title="E39">D</font></span></a>, <a class="txt" href="#E10:70_2_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="funct_1.html#T34" onmouseover="rs('funct_1/T34')" onmouseout="rh()">FUNCT_1:34</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K3" title="FIELD_1:func.3">PolyHom</a> <font color="Maroon" title="c19">iso</font></span>)</span> <a href="field_1.html#K2" title="FIELD_1:func.2">.</a> <span class="p2">(<span class="default"><a href="field_1.html#K6" title="FIELD_1:func.6">emb</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K3" title="FIELD_1:func.3">PolyHom</a> <font color="Maroon" title="c19">iso</font></span>)</span> <a href="field_1.html#K2" title="FIELD_1:func.2">.</a> <span class="p2">(<span class="default"><a href="field_1.html#K6" title="FIELD_1:func.6">emb</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:70_2_2"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K3" title="FIELD_1:func.3">PolyHom</a> <font color="Maroon" title="c19">iso</font></span>)</span> <a href="field_1.html#K2" title="FIELD_1:func.2">.</a> <span class="p2">(<span class="default"><a href="field_1.html#K6" title="FIELD_1:func.6">emb</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">i</font> = <font color="Maroon" title="c23">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K3" title="FIELD_1:func.3">PolyHom</a> <font color="Maroon" title="c19">iso</font></span>)</span> <a href="field_1.html#K2" title="FIELD_1:func.2">.</a> <span class="p2">(<span class="default"><a href="field_1.html#K6" title="FIELD_1:func.6">emb</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font> = 
<font color="Maroon" title="c19">iso</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K6" title="FIELD_1:func.6">emb</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_1.html#D2" onmouseover="rs('field_1/D2')" onmouseout="rh()">FIELD_1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c19">iso</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p2">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> (<span class="p3">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4"><a href="ring_4.html#K1" title="RING_4:func.1">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="ring_4.html#K1" title="RING_4:func.1">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_1.html#T40" onmouseover="rs('field_1/T40')" onmouseout="rh()">FIELD_1:40</a></span>
<br/>.= 
<font color="Maroon" title="c19">iso</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K5" title="FIELD_1:func.5">emb</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_1.html#T39" onmouseover="rs('field_1/T39')" onmouseout="rh()">FIELD_1:39</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:70_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_1.html#K3" title="FIELD_1:func.3">PolyHom</a> <font color="Maroon" title="c19">iso</font></span>)</span> <a href="field_1.html#K2" title="FIELD_1:func.2">.</a> <span class="p2">(<span class="default"><a href="field_1.html#K6" title="FIELD_1:func.6">emb</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_2"><span class="lab"><font color="Green" title="E37">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>
<span class="kw">hence </span><a NAME="E3:70_2"/>
<span class="p1">(<span class="default"><a href="field_1.html#K3" title="FIELD_1:func.3">PolyHom</a> <font color="Maroon" title="c19">iso</font></span>)</span> <a href="field_1.html#K2" title="FIELD_1:func.2">.</a> <span class="p1">(<span class="default"><a href="field_1.html#K6" title="FIELD_1:func.6">emb</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E28:70"/><span class="kw">then </span>
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c21">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="c20">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:70"><span class="lab"><font color="Green" title="E36">Z</font></span></a>, <a class="ref" href="field_4.html#T26" onmouseover="rs('field_4/T26')" onmouseout="rh()">FIELD_4:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:70"/>
<font color="Maroon" title="c2">p</font> <a href="field_4.html#R2" title="FIELD_4:pred.2">is_with_roots_in</a> <font color="Maroon" title="c20">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D3" onmouseover="rs('field_4/D3')" onmouseout="rh()">FIELD_4:def 3</a>, <a class="ref" href="field_4.html#D2" onmouseover="rs('field_4/D2')" onmouseout="rh()">FIELD_4:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
