<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">R</font> =  <a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">I</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> </span> } </span> ;<br/>
<a NAME="E1:89_1_1"/><span class="lab"><font color="Green" title="E65">A2</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="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="fvaluat1.html#K6" title="FVALUAT1:func.6">NonNegElements</a> <font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="fvaluat1.html#D12" target="_self" title="FVALUAT1:def.12">Def14</a></span>;<br/>
<a NAME="E2:89_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_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> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> </span> } </span>  or <font color="Maroon" title="c5">a</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 <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><br/>

<span class="kw">assume </span><a NAME="E1:89_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</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>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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><br/>

<a NAME="E2:89_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> st <br/>( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:89_1_1_1"/>
<font color="Maroon" title="c5">a</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 <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="#E1:89_1_1"><span class="lab"><font color="Green" title="E65">A2</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">I</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> </span> } </span>  as   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 <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="#E1:89"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="fvaluat1.html#E82"><span class="lab"><font color="Green" title="E63">Lm10</font></span></a></span>;<br/>
<a NAME="E4:89_1_1"/><span class="lab"><font color="Green" title="E66">A3</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="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_1_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="fvaluat1.html#T48" target="_self" title="FVALUAT1:th.48">Th51</a></span>;<br/>
<a NAME="E5:89_1_1"/>
( <font color="Maroon" title="c5">I</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>  &amp; <font color="Maroon" title="c5">I</font> is  <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a>  &amp; <font color="Maroon" title="c5">I</font> is  <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a> <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">I</font> is  <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a>  &amp; <font color="Maroon" title="c5">I</font> is  <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font> implies <font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font> )</span><br/><span class="kw">assume </span><a NAME="E1:89_1_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">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="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font> implies <font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E3:89_1_1_2_1"/><span class="lab"><font color="Green" title="E67">A4</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E4:89_1_1_2_1"/><span class="lab"><font color="Green" title="E68">A5</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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">x1</font>
 ;<br/><span class="kw">assume </span><a NAME="E5:89_1_1_2_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">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="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E7:89_1_1_2_1"/><span class="lab"><font color="Green" title="E69">A6</font></span>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">and </span><br/><a NAME="E8:89_1_1_2_1"/><span class="lab"><font color="Green" title="E70">A7</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c9">y1</font>
 ;<br/><a NAME="E9:89_1_1_2_1"/><span class="lab"><font color="Green" title="E71">A8</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E3:89_1_1_2_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E7:89_1_1_2_1"><span class="lab"><font color="Green" title="E69">A6</font></span></a>, <a class="ref" href="fvaluat1.html#T54" target="_self" title="FVALUAT1:th.54">Th57</a></span>;<br/><a NAME="E10:89_1_1_2_1"/><span class="lab"><font color="Green" title="E72">A9</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</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">x1</font></span>)</span>,<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="c9">y1</font></span>)</span>) <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">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="fvaluat1.html#T27" target="_self" title="FVALUAT1:th.27">Th28</a></span>;<br/><a NAME="E11:89_1_1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</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">x1</font></span>)</span>,<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="c9">y1</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_1_1_2_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E8:89_1_1_2_1"><span class="lab"><font color="Green" title="E70">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T21" title="XXREAL_0:th.21">XXREAL_0:21</a></span>;<br/><span class="kw">hence </span><a NAME="E12:89_1_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:89_1_1_2_1"><span class="lab"><font color="Green" title="E71">A8</font></span></a>, <a class="txt" href="#E10:89_1_1_2_1"><span class="lab"><font color="Green" title="E72">A9</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><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a> <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">I</font> is  <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font> implies <font color="Maroon" title="c6">p</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font> )</span><br/><span class="kw">assume </span><a NAME="E1:89_1_1_2_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">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="c6">p</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E3:89_1_1_2_2"/><span class="lab"><font color="Green" title="E67">A10</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E4:89_1_1_2_2"/><span class="lab"><font color="Green" title="E68">A11</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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">x1</font>
 ;<br/><a NAME="E5:89_1_1_2_2"/>
<font color="Maroon" title="c6">p</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 <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>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">p1</font> = <font color="Maroon" title="c6">p</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="#E4:89_1_1"><span class="lab"><font color="Green" title="E66">A3</font></span></a></span>;<br/><a NAME="E7:89_1_1_2_2"/><span class="lab"><font color="Green" title="E69">A12</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E3:89_1_1_2_2"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="fvaluat1.html#T55" target="_self" title="FVALUAT1:th.55">Th58</a></span>;<br/><a NAME="E8:89_1_1_2_2"/><span class="lab"><font color="Green" title="E70">A13</font></span>: 
<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="c9">p1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c9">p1</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="fvaluat1.html#D8" target="_self" title="FVALUAT1:def.8">Def10</a></span>;<br/><a NAME="E9:89_1_1_2_2"/>
 <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> <font color="Maroon" title="c9">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_1_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="fvaluat1.html#T47" target="_self" title="FVALUAT1:th.47">Th50</a></span>;<br/><span class="kw">hence </span><a NAME="E10:89_1_1_2_2"/>
<font color="Maroon" title="c6">p</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_1_1_2_2"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E7:89_1_1_2_2"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="txt" href="#E8:89_1_1_2_2"><span class="lab"><font color="Green" title="E70">A13</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>

<span class="kw">let </span><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ideal_1.html#D3" title="IDEAL_1:def.3">IDEAL_1: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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font> or <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:89_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">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="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E5:89_1_1_2"/><span class="lab"><font color="Green" title="E67">A14</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E6:89_1_1_2"/><span class="lab"><font color="Green" title="E68">A15</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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">x1</font>
 ;<br/>
<a NAME="E7:89_1_1_2"/>
<font color="Maroon" title="c6">p</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 <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>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">p1</font> = <font color="Maroon" title="c6">p</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="#E4:89_1_1"><span class="lab"><font color="Green" title="E66">A3</font></span></a></span>;<br/>
<a NAME="E9:89_1_1_2"/><span class="lab"><font color="Green" title="E69">A16</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E5:89_1_1_2"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="ref" href="fvaluat1.html#T55" target="_self" title="FVALUAT1:th.55">Th58</a></span>;<br/>
<a NAME="E10:89_1_1_2"/><span class="lab"><font color="Green" title="E70">A17</font></span>: 
<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="c9">p1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">x1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c9">p1</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</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">x1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="fvaluat1.html#D8" target="_self" title="FVALUAT1:def.8">Def10</a></span>;<br/>
<a NAME="E11:89_1_1_2"/>
 <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> <font color="Maroon" title="c9">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_1_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="fvaluat1.html#T47" target="_self" title="FVALUAT1:th.47">Th50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:89_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:89_1_1_2"><span class="lab"><font color="Green" title="E68">A15</font></span></a>, <a class="txt" href="#E9:89_1_1_2"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="txt" href="#E10:89_1_1_2"><span class="lab"><font color="Green" title="E70">A17</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>
<span class="kw">hence </span><a NAME="E6:89_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> </span> } </span>  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"> verum</span><br/>


</div>
