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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/> <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Olive" title="b1">v</font> is  <a href="ring_1.html#V4" title="RING_1:attr.4">maximal</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  <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font> is  <a href="ring_1.html#V4" title="RING_1:attr.4">maximal</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:116"/><span class="lab"><font color="Green" title="E77">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">  <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font> is  <a href="ring_1.html#V4" title="RING_1:attr.4">maximal</a> </span><br/>

<span class="kw">thus </span><a NAME="E2:116"/>
 <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ring_1.html#D4" title="RING_1:def.4">RING_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font> is <a href="ring_1.html#V3" title="RING_1:attr.3">V247</a>( <a href="fvaluat1.html#K7" title="FVALUAT1:func.7">ValuatRing</a> <font color="Maroon" title="c2">v</font>)</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:116_1"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</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#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="fvaluat1.html#D12" target="_self" title="FVALUAT1:def.12">Def14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:116_1"/>
 <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="#E1:116"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="fvaluat1.html#T63" target="_self" title="FVALUAT1:th.63">Th66</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="subset_1.html#D6" title="SUBSET_1:def.6">SUBSET_1:def 6</a> <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="c3">J</font> be   <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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ring_1.html#D3" title="RING_1:def.3">RING_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  <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">J</font> or <font color="Maroon" title="c3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font> or  not <font color="Maroon" title="c3">J</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:116"/><span class="lab"><font color="Green" title="E78">A2</font></span>: 
 <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">J</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">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font> or  not <font color="Maroon" title="c3">J</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )</span><br/>

<a NAME="E4:116"/>
(  not <font color="Maroon" title="c3">J</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  or <font color="Maroon" title="c3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:116_2"/>
<font color="Maroon" title="c3">J</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</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="c3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font></span><br/>

<a NAME="E2:116_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="fvaluat1.html#T79" target="_self" title="FVALUAT1:th.79">Th82</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:116_2"/>
<font color="Maroon" title="c3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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="E5:116"/>
( <font color="Maroon" title="c3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvaluat1.html#NK10" title="FVALUAT1:NK.10">vp</a> <font color="Maroon" title="c2">v</font> or  not <font color="Maroon" title="c3">J</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )
 ; <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>
