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<b>let </b><font color="Maroon" title="c1">R</font> be  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b1">I</font> is <a href="ring_1.html#V4" title="RING_1:attr.4">maximal</a> iff <font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Olive" title="b1">I</font> is   <a href="vectsp_2.html#NM3" title="VECTSP_2:NM.3">Skew-Field</a> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V4" title="RING_1:attr.4">maximal</a> iff <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_2.html#NM3" title="VECTSP_2:NM.3">Skew-Field</a> )</span><br/>



<b>thus </b><a NAME="E1:38"/>
( <font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V4" title="RING_1:attr.4">maximal</a> implies <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_2.html#NM3" title="VECTSP_2:NM.3">Skew-Field</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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_2.html#NM3" title="VECTSP_2:NM.3">Skew-Field</a> implies <font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V4" title="RING_1:attr.4">maximal</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V4" title="RING_1:attr.4">maximal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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_2.html#NM3" title="VECTSP_2:NM.3">Skew-Field</a></span><br/>

<a NAME="E2:38_1"/><b>then </b>
( <font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V3" title="RING_1:attr.3">quasi-maximal</a> &amp; <font color="Maroon" title="c2">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> )
 
<b>by </b><i><a class="ref" href="ring_1.html#D4" target="_self" title="RING_1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1"/>
<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_2.html#NM3" title="VECTSP_2:NM.3">Skew-Field</a>
 <b>by </b><i><a class="ref" href="ring_1.html#T16" target="_self" title="RING_1:th.16">Th16</a>, <a class="ref" href="ring_1.html#T19" target="_self" title="RING_1:th.19">Th19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:38"/>
<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_2.html#NM3" title="VECTSP_2:NM.3">Skew-Field</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V4" title="RING_1:attr.4">maximal</a></span><br/>

<b>hence </b><a NAME="E3:38"/>
( <font color="Maroon" title="c2">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> &amp; <font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V3" title="RING_1:attr.3">quasi-maximal</a> )
 <b>by </b><i><a class="ref" href="ring_1.html#T16" target="_self" title="RING_1:th.16">Th16</a>, <a class="ref" href="ring_1.html#T20" target="_self" title="RING_1:th.20">Th20</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ring_1.html#D4" target="_self" title="RING_1:def.4">RING_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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