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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="algstr_0.html#V7" title="ALGSTR_0:attr.7">add-cancelable</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</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">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b1">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  &amp; <font color="Olive" title="b1">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="ideal_1.html#K9" title="IDEAL_1:func.9">-RightIdeal</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">F</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  &amp; <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K9" title="IDEAL_1:func.9">-RightIdeal</a>  )</span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  ) &amp; ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  )</span>
<div class="add"><b>assume </b><a NAME="E1:90_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> </span><br/><b>then consider </b><font color="Maroon" title="c4">lc</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E3:90_1_1"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c4">lc</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T60" target="_self" title="IDEAL_1:th.60">Th60</a></i>;<br/><a NAME="E4:90_1_1"/>
<font color="Maroon" title="c4">lc</font> is    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T31" target="_self" title="IDEAL_1:th.31">Th31</a></i>;<br/><b>hence </b><a NAME="E5:90_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 <b>by </b><i><a class="txt" href="#E3:90_1_1"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="ideal_1.html#T61" target="_self" title="IDEAL_1:th.61">Th61</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="E2:90_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/><b>then consider </b><font color="Maroon" title="c4">lc</font> being    <a href="ideal_1.html#M2" title="IDEAL_1:mode.2">LeftLinearCombination</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E4:90_1"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c4">lc</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T61" target="_self" title="IDEAL_1:th.61">Th61</a></i>;<br/><a NAME="E5:90_1"/>
<font color="Maroon" title="c4">lc</font> is    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T25" target="_self" title="IDEAL_1:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E6:90_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="#E4:90_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="ideal_1.html#T60" target="_self" title="IDEAL_1:th.60">Th60</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="E2:90"/>
<font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K8" title="IDEAL_1:func.8">-LeftIdeal</a> 
 <b>by </b><i><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"> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K9" title="IDEAL_1:func.9">-RightIdeal</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K9" title="IDEAL_1:func.9">-RightIdeal</a>  ) &amp; ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K9" title="IDEAL_1:func.9">-RightIdeal</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K9" title="IDEAL_1:func.9">-RightIdeal</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  )</span>
<div class="add"><b>assume </b><a NAME="E1:90_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K9" title="IDEAL_1:func.9">-RightIdeal</a> </span><br/><b>then consider </b><font color="Maroon" title="c4">lc</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E3:90_2_1"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c4">lc</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T60" target="_self" title="IDEAL_1:th.60">Th60</a></i>;<br/><a NAME="E4:90_2_1"/>
<font color="Maroon" title="c4">lc</font> is    <a href="ideal_1.html#M3" title="IDEAL_1:mode.3">RightLinearCombination</a> of <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T31" target="_self" title="IDEAL_1:th.31">Th31</a></i>;<br/><b>hence </b><a NAME="E5:90_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K9" title="IDEAL_1:func.9">-RightIdeal</a> 
 <b>by </b><i><a class="txt" href="#E3:90_2_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="ideal_1.html#T62" target="_self" title="IDEAL_1:th.62">Th62</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="E2:90_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K9" title="IDEAL_1:func.9">-RightIdeal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/><b>then consider </b><font color="Maroon" title="c4">lc</font> being    <a href="ideal_1.html#M3" title="IDEAL_1:mode.3">RightLinearCombination</a> of <font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E4:90_2"/><i><font color="Green" title="E45">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c4">lc</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T62" target="_self" title="IDEAL_1:th.62">Th62</a></i>;<br/><a NAME="E5:90_2"/>
<font color="Maroon" title="c4">lc</font> is    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T28" target="_self" title="IDEAL_1:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E6:90_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="#E4:90_2"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="ideal_1.html#T60" target="_self" title="IDEAL_1:th.60">Th60</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="E4:90"/>
<font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="ideal_1.html#K9" title="IDEAL_1:func.9">-RightIdeal</a> 
 <b>by </b><i><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/>


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