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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="algstr_1.html#V1">left_zeroed</a> <a href="binom.html#V1">add-cancelable</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>



<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:116_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 ;<br/><a NAME="E2:116_1"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Olive">r</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T3" target="_self">Th3</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:116_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font></span>)</span>
 ;<br/><a NAME="E5:116_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span> 
 ;<br/><b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font> as    <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <b>by </b><i/>;<br/><a NAME="E7:116_1"/>
<font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span> 
 ;<br/><b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font> as    <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <b>by </b><i/>;<br/><a NAME="E9:116_1"/>
<font color="Maroon">a'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E10:116_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> </span>)</span> <a href="ideal_1.html#K12" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> </span>)</span>
 <b>by </b><i><a class="ref" href="ideal_1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:116_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> </span>)</span> <a href="ideal_1.html#K12" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> </span>)</span>
 ;<br/><a NAME="E2:116_2"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:116_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  )
 ;<br/><a NAME="E5:116_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T11" target="_self">Th11</a>, </i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E7:116_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E8:116_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T11" target="_self">Th11</a>, </i>;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E10:116_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E11:116_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Olive">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Olive">d</font></span>)</span></span>)</span> where <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T11" target="_self">Th11</a>, <a class="ref" href="ideal_1.html#T12" target="_self">Th12</a>, </i>;<br/><b>hence </b><a NAME="E12:116_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:116"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> </span>)</span> <a href="ideal_1.html#K12" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="ideal_1.html#K7" target="_self">-Ideal</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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