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<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">I</font> be   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>set </b><font color="Maroon" title="c3">E</font> =  <a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font>;<br/>
<a NAME="E1:35"/><i><font color="Green" title="E23">A1</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ring_1.html#D6" target="_self" title="RING_1:def.6">Def6</a></i>;<br/>
<a NAME="E2:35"/><i><font color="Green" title="E24">A2</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ring_1.html#T15" target="_self" title="RING_1:th.15">Th15</a></i>;<br/>
<a NAME="E3:35"/><i><font color="Green" title="E25">A3</font></i>: 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>;<br/>
<b>thus </b><a NAME="E4:35"/>
( <font color="Maroon" title="c2">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> implies  not <font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font> is <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:35_1"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c2">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 <b>and </b><br/><a NAME="E2:35_1"/><i><font color="Green" title="E27">A5</font></i>: 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D21" title="VECTSP_1:def.21">VECTSP_1:def 21</a><br/>

<a NAME="E3:35_1"/>
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E2:35_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="ring_1.html#E1:35"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="ring_1.html#E2:35"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="ring_1.html#T6" target="_self" title="RING_1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:35_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ring_1.html#E1:35_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="ring_1.html#E3:35"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="ideal_1.html#T19" title="IDEAL_1:th.19">IDEAL_1:19</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:35"/><i><font color="Green" title="E26">A6</font></i>: 
not <font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font> is <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a>
 ;<br/>

<b>assume </b><a NAME="E6:35"/>
not <font color="Maroon" title="c2">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 ;<br/>

<a NAME="E7:35"/><b>then </b>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T19" title="IDEAL_1:th.19">IDEAL_1:19</a></i>;<br/>
<a NAME="E8:35"/><b>then </b>
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ring_1.html#E3:35"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="ring_1.html#T6" target="_self" title="RING_1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E9:35"/>
contradiction
 <b>by </b><i><a class="txt" href="ring_1.html#E1:35"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="ring_1.html#E2:35"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="ring_1.html#E5:35"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="vectsp_1.html#D21" title="VECTSP_1:def.21">VECTSP_1:def 21</a></i>;<br/>


</div>
