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<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a>  <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#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">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#R2" title="FIELD_4:pred.2">is_with_roots_in</a> <font color="Olive" title="b1">E</font></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">p</font> being    <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"> such that </span><br/><a NAME="E2:77"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="field_1.html#R1" title="FIELD_1:pred.1">is_a_irreducible_factor_of</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_1.html#T3" onmouseover="rs('field_1/T3')" onmouseout="rh()">FIELD_1:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c3">p</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="field_1.html#D1" onmouseover="rs('field_1/D1')" onmouseout="rh()">FIELD_1:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">q</font> being    <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"> such that </span><br/><a NAME="E5:77"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="field_1.html#D1" onmouseover="rs('field_1/D1')" onmouseout="rh()">FIELD_1:def 1</a>, <a class="ref" href="gcd_1.html#D1" onmouseover="rs('gcd_1/D1')" onmouseout="rh()">GCD_1:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">E</font> being   <a href="field_4.html#NM4" title="FIELD_4:NM.4">FieldExtension</a> of <font color="Maroon" title="c1">F</font><span class="kw"> such that </span><br/><a NAME="E7:77"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="field_4.html#R2" title="FIELD_4:pred.2">is_with_roots_in</a> <font color="Maroon" title="c6">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="field_4.html#E51"><span class="lab"><font color="Green" title="E30">Lm11</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">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">f</font> <a href="field_4.html#R2" title="FIELD_4:pred.2">is_with_roots_in</a> <font color="Maroon" title="c6">E</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">E</font><span class="kw"> such that </span><br/><a NAME="E9:77"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="field_4.html#R1" title="FIELD_4:pred.1">is_a_root_of</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:77"><span class="lab"><font color="Green" title="E33">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">p1</font> = <font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c9">q1</font> = <font color="Maroon" title="c5">q</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font> ;<br/>
<a NAME="E11:77"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
<font color="Maroon" title="c1">F</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Maroon" title="c6">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_4.html#D1" target="_self" onmouseover="rs('field_4/D1')" onmouseout="rh()">Def1</a></span>;<br/>
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">a</font>) = 
 <a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">p1</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c9">q1</font></span>)</span>,<font color="Maroon" title="c7">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="polynom3.html#D10" onmouseover="rs('polynom3/D10')" onmouseout="rh()">POLYNOM3:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c7">a</font>)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="algnum_1.html#K1" title="ALGNUM_1:func.1">Ext_eval</a> (<font color="Maroon" title="c9">q1</font>,<font color="Maroon" title="c7">a</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:77"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="algnum_1.html#T20" onmouseover="rs('algnum_1/T20')" onmouseout="rh()">ALGNUM_1:20</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">E</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:77"><span class="lab"><font color="Green" title="E34">A4</font></span></a></span>
;<br/>
<a NAME="E13:77"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="field_4.html#R1" title="FIELD_4:pred.1">is_a_root_of</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">E</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E14:77"/>
<font color="Maroon" title="c2">f</font> <a href="field_4.html#R2" title="FIELD_4:pred.2">is_with_roots_in</a> <font color="Maroon" title="c6">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"> verum</span><br/>


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