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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">I</font>, <font color="Maroon" title="c5">J</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c4">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c5">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">I</font></span><br/><div><i><font color="Green" title="E51">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c5">J</font> implies <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">I</font> )</span><br/><b>assume </b><a NAME="E1:107_1_1_1_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c5">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">I</font></span><br/><b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:107_1_1_1_1"/><i><font color="Green" title="E52">A2</font></i>: 
( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> )
 ;<br/><b>thus </b><a NAME="E4:107_1_1_1_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:107_1_1_1_1"><i><font color="Green" title="E52">A2</font></i></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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">I</font> implies <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c5">J</font> )</span><br/><b>assume </b><a NAME="E1:107_1_1_1_2"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c5">J</font></span><br/><b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:107_1_1_1_2"/><i><font color="Green" title="E52">A3</font></i>: 
( <font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> )
 ;<br/><b>thus </b><a NAME="E4:107_1_1_1_2"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c5">J</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:107_1_1_1_2"><i><font color="Green" title="E52">A3</font></i></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>hence </b><a NAME="E3:107_1_1_1"/>
<font color="Maroon" title="c4">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c5">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Maroon" title="c4">I</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:107_1_1_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</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>hence </b><a NAME="E2:107_1_1"/>
for <font color="Olive" title="b1">I</font>, <font color="Olive" title="b2">J</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> holds  <font color="Olive" title="b1">I</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Olive" title="b2">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">J</font> <a href="ideal_1.html#K11" title="IDEAL_1:func.11">+</a> <font color="Olive" title="b1">I</font>
 ; <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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