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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">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> holds <br/> (  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Olive" title="b1">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a>  iff <font color="Olive" title="b1">p</font> is  <a href="field_5.html#V1" title="FIELD_5:attr.1">linear</a>  )</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"> (  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a>  iff <font color="Maroon" title="c2">p</font> is  <a href="field_5.html#V1" title="FIELD_5:attr.1">linear</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">FP</font> =  <a href="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">K</font> =  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">X</font> = <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>;<br/>
<a NAME="E1:40"/><span class="lab"><font color="Green" title="E18">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> <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> <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="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font></span>)</span> )
 
;<br/>
<div><span class="lab"><font color="Green" title="E19">A</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 implies  not  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E20">AS</font></span>: 
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a> </span><br/><a NAME="E2:40_1"/><span class="lab"><font color="Green" title="E21">H</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">F</font> :  <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <font color="Olive" title="b1">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c2">p</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ring_4.html#D8" onmouseover="rs('ring_4/D8')" onmouseout="rh()">RING_4:def 8</a></span>;<br/><a NAME="E3:40_1"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T40" onmouseover="rs('polynom5/T40')" onmouseout="rh()">POLYNOM5:40</a></span>;<br/><a NAME="E4:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">D</font></span>: 
 <a href="hurwitz.html#NK3" title="HURWITZ:NK.3">deg</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hurwitz.html#D2" onmouseover="rs('hurwitz/D2')" onmouseout="rh()">HURWITZ:def 2</a></span>;<br/><a NAME="E5:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">C</font></span>: 
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <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="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="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E21">H</font></span></a>, <a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E20">AS</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_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 <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_1"/>
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:40_1_1"/><span class="lab"><font color="Green" title="E24">B1</font></span>: 
( <font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">o</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">F</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_1"><span class="lab"><font color="Green" title="E24">B1</font></span></a></span>;<br/><a NAME="E5:40_1_1"/>
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">a</font> <a href="ring_4.html#K2" title="RING_4:func.2">|</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_1"><span class="lab"><font color="Green" title="E24">B1</font></span></a>, <a class="ref" href="field_5.html#D2" target="_self" onmouseover="rs('field_5/D2')" onmouseout="rh()">defch</a></span>;<br/><span class="kw">hence </span><a NAME="E6:40_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1"><span class="lab"><font color="Green" title="E22">D</font></span></a>, <a class="ref" href="ratfunc1.html#D2" onmouseover="rs('ratfunc1/D2')" onmouseout="rh()">RATFUNC1: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><span class="kw">end;</span></div><a NAME="E7:40_1"/><span class="kw">then </span>
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <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="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E23">C</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E8:40_1"/><span class="kw">then </span>
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="ring_4.html#K6" title="RING_4:func.6">Polynom-Ring</a> <font color="Maroon" title="c2">p</font></span>)</span> <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> <span class="p3">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></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>
 <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="E9:40_1"/><span class="kw">then </span>
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="field_2.html#K2" title="FIELD_2:func.2">carr</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E18">X</font></span></a>, <a class="ref" href="field_2.html#D2" onmouseover="rs('field_2/D2')" onmouseout="rh()">FIELD_2:def 2</a></span>;<br/><a NAME="E10:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A</font></span>: 
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <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_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
 <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/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">n</font> be    <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> ; <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"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_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="E1:40_1_2_1"/>
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T23" onmouseover="rs('nat_1/T23')" onmouseout="rh()">NAT_1:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_1"/><span class="lab"><font color="Green" title="E25">B</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
<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"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_2_1_1"><span class="lab"><font color="Green" title="E25">B</font></span></a>, <a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/> <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="40_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_2"/><span class="lab"><font color="Green" title="E25">B</font></span>: 
<font color="Maroon" title="c6">n</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"> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font> = 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
<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"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_2_1_2"><span class="lab"><font color="Green" title="E25">B</font></span></a>, <a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/> <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="40_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:40_1_2_1_3"/><span class="lab"><font color="Green" title="E25">B</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
<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"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_2_1_3"><span class="lab"><font color="Green" title="E25">B</font></span></a>, <a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>
;<br/> <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><a NAME="E12:40_1"/><span class="kw">then </span>
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>
 ;<br/><a NAME="E13:40_1"/><span class="kw">then </span>
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <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="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></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="E14:40_1"/><span class="kw">then </span>
<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1"><span class="lab"><font color="Green" title="E24">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/><span class="kw">hence </span><a NAME="E15:40_1"/>
 not  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_3.html#D3" onmouseover="rs('field_3/D3')" onmouseout="rh()">FIELD_3:def 3</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>
<a NAME="E3:40"/><span class="lab"><font color="Green" title="E20">H</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</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="c1">F</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/>
<div><span class="lab"><font color="Green" title="E21">B</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="40_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"> ( <font color="Maroon" title="c2">p</font> is  <a href="field_5.html#V1" title="FIELD_5:attr.1">linear</a>  implies  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_2"/><span class="lab"><font color="Green" title="E22">C</font></span>: 
<font color="Maroon" title="c2">p</font> is  <a href="field_5.html#V1" title="FIELD_5:attr.1">linear</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a> </span><br/><a NAME="E2:40_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">D</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <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="#E3:40"><span class="lab"><font color="Green" title="E20">H</font></span></a>, <a class="txt" href="field_5.html#E17"><span class="lab"><font color="Green" title="E12">lempk</font></span></a>, <a class="ref" href="field_5.html#T18" target="_self" onmouseover="rs('field_5/T18')" onmouseout="rh()">polyd</a></span>;<br/><a NAME="E3:40_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p2">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</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#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_2"><span class="lab"><font color="Green" title="E22">C</font></span></a>, <a class="ref" href="field_5.html#T18" target="_self" onmouseover="rs('field_5/T18')" onmouseout="rh()">polyd</a></span>;<br/><a NAME="E4:40_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p3">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">Polynom-Ring</a> <span class="p3">(<span class="default"><a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p4">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span></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="txt" href="#E2:40_2"><span class="lab"><font color="Green" title="E23">D</font></span></a>, <a class="ref" href="field_3.html#D3" onmouseover="rs('field_3/D3')" onmouseout="rh()">FIELD_3:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:40_2"/>
 <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_3.html#D3" onmouseover="rs('field_3/D3')" onmouseout="rh()">FIELD_3:def 3</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="40_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a>  implies <font color="Maroon" title="c2">p</font> is  <a href="field_5.html#V1" title="FIELD_5:attr.1">linear</a>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_3"/><span class="lab"><font color="Green" title="E22">C</font></span>: 
 <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</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="c2">p</font> is  <a href="field_5.html#V1" title="FIELD_5:attr.1">linear</a> </span><br/><a NAME="E2:40_3"/>
 <a href="field_1.html#K1" title="FIELD_1:func.1">deg</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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="ref" href="int_1.html#T7" onmouseover="rs('int_1/T7')" onmouseout="rh()">INT_1:7</a>, <a class="ref" href="ring_4.html#D4" onmouseover="rs('ring_4/D4')" onmouseout="rh()">RING_4:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_3"/>
<font color="Maroon" title="c2">p</font> is  <a href="field_5.html#V1" title="FIELD_5:attr.1">linear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E19">A</font></span></a>, <a class="txt" href="#E1:40_3"><span class="lab"><font color="Green" title="E22">C</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</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">hence </span><a NAME="E6:40"/>
(  <a href="field_2.html#K7" title="FIELD_2:func.7">embField</a> <span class="p1">(<span class="default"><a href="field_5.html#K1" title="FIELD_5:func.1">canHomP</a> <font color="Maroon" title="c2">p</font></span>)</span> is  <a href="field_3.html#V3" title="FIELD_3:attr.3">polynomial_disjoint</a>  iff <font color="Maroon" title="c2">p</font> is  <a href="field_5.html#V1" title="FIELD_5:attr.1">linear</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E21">B</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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