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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="vectsp_2.html#NM2" title="VECTSP_2:NM.2">domRing</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> holds   <a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Olive" title="b2">q</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c3">q</font></span>)</span></span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> implies <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:81_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
 <a href="uproots.html#K7" title="UPROOTS:func.7">multiplicity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D9" target="_self" title="UPROOTS:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K7" title="UPROOTS:func.7">multiplicity</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K7" title="UPROOTS:func.7">multiplicity</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T57" target="_self" title="UPROOTS:th.57">Th57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K7" title="UPROOTS:func.7">multiplicity</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D9" target="_self" title="UPROOTS:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D9" target="_self" title="UPROOTS:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:81"/>
 <a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" title="UPROOTS:func.8">BRoots</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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