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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="realset2.html#V1" title="REALSET2:attr.1">Field-like</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="fvaluat1.html#M1" title="FVALUAT1:mode.1">Valuation</a> of <font color="Maroon" title="c1">K</font>  st <font color="Maroon" title="c1">K</font> is  <a href="fvaluat1.html#V3" title="FVALUAT1:attr.3">having_valuation</a>  holds <br/> for <font color="Olive" title="b2">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Olive" title="b1">v</font></span>)</span>  st <font color="Olive" title="b2">I</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Olive" title="b1">v</font></span>)</span> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fvaluat1.html#K9" title="FVALUAT1:func.9">min</a> (<font color="Olive" title="b2">I</font>,<font color="Olive" title="b1">v</font>) holds <br/><font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font> be    <a href="fvaluat1.html#M1" title="FVALUAT1:mode.1">Valuation</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">K</font> is  <a href="fvaluat1.html#V3" title="FVALUAT1:attr.3">having_valuation</a>  implies  for <font color="Olive" title="b1">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span>  st <font color="Olive" title="b1">I</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fvaluat1.html#K9" title="FVALUAT1:func.9">min</a> (<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">v</font>) holds <br/><font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:97"/><span class="lab"><font color="Green" title="E70">A1</font></span>: 
<font color="Maroon" title="c1">K</font> is  <a href="fvaluat1.html#V3" title="FVALUAT1:attr.3">having_valuation</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span>  st <font color="Olive" title="b1">I</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fvaluat1.html#K9" title="FVALUAT1:func.9">min</a> (<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c2">v</font>) holds <br/><font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">I</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" 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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span>  st <font color="Maroon" title="c3">I</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fvaluat1.html#K9" title="FVALUAT1:func.9">min</a> (<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">v</font>) holds <br/><font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">I</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fvaluat1.html#K9" title="FVALUAT1:func.9">min</a> (<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">v</font>) implies <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E2:97"/><span class="lab"><font color="Green" title="E71">A2</font></span>: 
<font color="Maroon" title="c3">I</font> is   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fvaluat1.html#K9" title="FVALUAT1:func.9">min</a> (<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">v</font>) or <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:97"/><span class="lab"><font color="Green" title="E72">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fvaluat1.html#K9" title="FVALUAT1:func.9">min</a> (<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">v</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/>

<a NAME="E4:97"/><span class="lab"><font color="Green" title="E73">A4</font></span>: 
 <a href="fvaluat1.html#K9" title="FVALUAT1:func.9">min</a> (<font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c2">v</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="fvaluat1.html#T64" target="_self" title="FVALUAT1:th.64">Th67</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:97"/>
<font color="Maroon" title="c3">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">I</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:97_1"/><span class="lab"><font color="Green" title="E74">A5</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">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="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E71">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c6">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="ref" href="fvaluat1.html#T51" target="_self" title="FVALUAT1:th.51">Th54</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:97_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="ref" href="fvaluat1.html#D12" target="_self" title="FVALUAT1:def.12">Def14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:97_1_1_1"/><span class="lab"><font color="Green" title="E75">A6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">r1</font> = <font color="Maroon" title="c8">y1</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">/</a> <font color="Maroon" title="c7">x1</font>;<br/><a NAME="E2:97_1_1_1"/>
<font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="txt" href="#E1:97_1"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="ref" href="fvaluat1.html#T65" target="_self" title="FVALUAT1:th.65">Th68</a></span>;<br/><a NAME="E3:97_1_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y1</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fvaluat1.html#E44"><span class="lab"><font color="Green" title="E36">Lm6</font></span></a></span>;<br/><a NAME="E4:97_1_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">/</a> <font color="Maroon" title="c7">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_1_1_1"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="ref" href="fvaluat1.html#T22" target="_self" title="FVALUAT1:th.22">Th23</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">r0</font> = <font color="Maroon" title="c8">y1</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">/</a> <font color="Maroon" title="c7">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="ref" href="fvaluat1.html#T52" target="_self" title="FVALUAT1:th.52">Th55</a></span>;<br/><font color="Maroon" title="c7">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="vectsp_2.html#K1" title="VECTSP_2:func.1">/</a> <font color="Maroon" title="c7">x1</font></span>)</span> = 
<font color="Maroon" title="c8">y1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c8">y1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_1_1_1"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="ref" href="vectsp_2.html#T9" title="VECTSP_2:th.9">VECTSP_2:9</a></span>
<br/>.= 
<font color="Maroon" title="c8">y1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
;<br/><a NAME="E7:97_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A7</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">r0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="ref" href="fvaluat1.html#T55" target="_self" title="FVALUAT1:th.55">Th58</a></span>;<br/><a NAME="E8:97_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">r</font></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span> : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#T64" title="IDEAL_1:th.64">IDEAL_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E9:97_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:97_1_1_1"><span class="lab"><font color="Green" title="E76">A7</font></span></a></span>; <a class="txt" 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="97_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:97_1_1_2"/><span class="lab"><font color="Green" title="E75">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/><div class="add"><a NAME="E2:97_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A9</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</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="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#T47" title="IDEAL_1:th.47">IDEAL_1:47</a></span>;<br/><a NAME="E3:97_1_1_2"/><span class="lab"><font color="Green" title="E77">A10</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></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">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="ref" href="fvaluat1.html#D12" target="_self" title="FVALUAT1:def.12">Def14</a></span>;<br/><a NAME="E4:97_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A11</font></span>: 
<font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E1:97_1_1_2"><span class="lab"><font color="Green" title="E75">A8</font></span></a>, <a class="ref" href="fvaluat1.html#D8" target="_self" title="FVALUAT1:def.8">Def10</a></span>;<br/><a NAME="E5:97_1_1_2"/>
<font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E1:97_1"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="fvaluat1.html#T65" target="_self" title="FVALUAT1:th.65">Th68</a></span>;<br/><a NAME="E6:97_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E3:97_1_1_2"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="ref" href="fvaluat1.html#T18" target="_self" title="FVALUAT1:th.18">Th19</a>, <a class="txt" href="#E4:97_1_1_2"><span class="lab"><font color="Green" title="E78">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:97_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_1_1_2"><span class="lab"><font color="Green" title="E76">A9</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" 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="E6:97"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:97"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="ideal_1.html#D14" title="IDEAL_1:def.14">IDEAL_1:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
