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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">right_unital</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a> <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>;<br/>



<div><i><font color="Green" title="E19">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c2">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 ;<br/><a NAME="E2:44_1"/><i><font color="Green" title="E21">A3</font></i>: 
not  <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:44_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
 <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>
 ;<br/>

<a NAME="E2:44_1_1"/><i><font color="Green" title="E23">A5</font></i>: 
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> holds <br/><font color="Olive" title="b1">u</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 <font color="Maroon" title="c1">R</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:44_1_1_1"/>
<font color="Maroon" title="c3">u</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 <font color="Maroon" title="c1">R</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">u'</font> = <font color="Maroon" title="c3">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/><a NAME="E3:44_1_1_1"/>
<font color="Maroon" title="c4">u'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <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> <font color="Maroon" title="c4">u'</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_1"/>
<font color="Maroon" title="c3">u</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="txt" href="ideal_1.html#E1:44_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="ideal_1.html#D2" target="_self" title="IDEAL_1:def.2">Def2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:44_1_1"/><b>then </b>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E2:44_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E5:44_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:44_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="ideal_1.html#D5" target="_self" title="IDEAL_1:def.5">Def5</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:44_1"/>
for <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">u</font> is <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a> holds <br/> not <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>

<b>assume </b><a NAME="E1:44_1_2"/>
<font color="Maroon" title="c3">u</font> is <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>
 ;<br/>

<a NAME="E2:44_1_2"/><b>then </b>
<font color="Maroon" title="c3">u</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</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="gcd_1.html#D2" title="GCD_1:def.2">GCD_1:def 2</a></i>;<br/>
<a NAME="E3:44_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">b</font>
 
<b>by </b><i><a class="ref" href="gcd_1.html#D1" title="GCD_1:def.1">GCD_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:44_1_2"/>
not <font color="Maroon" title="c3">u</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="txt" href="ideal_1.html#E2:44_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="ideal_1.html#D3" target="_self" title="IDEAL_1:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_2"/><i><font color="Green" title="E20">A6</font></i>: 
for <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">u</font> is <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a> holds <br/> not <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/><a NAME="E2:44_2"/>
<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="group_1.html#K1" title="GROUP_1:func.1">*</a> <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="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="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/><a NAME="E3:44_2"/><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="gcd_1.html#R1" title="GCD_1:pred.1">divides</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="gcd_1.html#D1" title="GCD_1:def.1">GCD_1:def 1</a></i>;<br/><a NAME="E4:44_2"/><b>then </b>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">R</font> is <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>
 <b>by </b><i><a class="ref" href="gcd_1.html#D2" title="GCD_1:def.2">GCD_1:def 2</a></i>;<br/><a NAME="E5:44_2"/><b>then </b>
<font color="Maroon" title="c2">I</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 <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:44_2"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E6:44_2"/>
<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="ideal_1.html#D5" target="_self" title="IDEAL_1:def.5">Def5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:44"/>
( <font color="Maroon" title="c2">I</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a> iff for <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">u</font> is <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a> holds <br/> not <font color="Olive" title="b1">u</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="txt" href="ideal_1.html#E1:44"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>


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