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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#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</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="vectsp_1.html#V2" title="VECTSP_1:attr.2">left-distributive</a> <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</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">I</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font><br/> for <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><br/> for <font color="Olive" title="b3">K</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>  st <font color="Olive" title="b2">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">I</font> holds <br/><font color="Olive" title="b1">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Olive" title="b3">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Olive" title="b3">K</font></span>)</span></span><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#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</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"> for <font color="Olive" title="b1">J</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">K</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>  st <font color="Olive" title="b1">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font> holds <br/><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Olive" title="b2">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Olive" title="b2">K</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">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"> for <font color="Olive" title="b1">K</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>  st <font color="Maroon" title="c3">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font> holds <br/><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Olive" title="b1">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Olive" title="b1">K</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">K</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="c3">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font> implies <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c4">K</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:121"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c3">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">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="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c4">K</font></span>)</span></span><br/>

<div><i><font color="Green" title="E54">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c4">K</font></span>)</span> implies <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c4">K</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:121_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c4">K</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c4">K</font></span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c6">v</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="E3:121_1"/><i><font color="Green" title="E55">A3</font></i>: 
( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c4">K</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c8">k</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="E5:121_1"/><i><font color="Green" title="E56">A4</font></i>: 
( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:121_1"><i><font color="Green" title="E55">A3</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">j'</font> = <font color="Maroon" title="c7">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font> <b>by </b><i><a class="txt" href="ideal_1.html#E1:121"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="ideal_1.html#E5:121_1"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/><a NAME="E7:121_1"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">j'</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="ref" href="ideal_1.html#T13" target="_self" title="IDEAL_1:th.13">Th13</a></i>;<br/><a NAME="E8:121_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">j'</font></span>)</span> <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#E3:121_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="ideal_1.html#E5:121_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">Def1</a></i>;<br/><a NAME="E9:121_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c9">j'</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">k</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="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/><a NAME="E10:121_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">k</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="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>;<br/><a NAME="E11:121_1"/><b>then </b>
<font color="Maroon" title="c8">k</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="ref" href="algstr_1.html#D5" title="ALGSTR_1:def.5">ALGSTR_1:def 5</a></i>;<br/><a NAME="E12:121_1"/><b>then </b>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c4">K</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E5:121_1"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E13:121_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c4">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:121_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="ideal_1.html#E5:121_1"><i><font color="Green" title="E56">A4</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="121_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c4">K</font></span>)</span> implies <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c4">K</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:121_2"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c4">K</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c4">K</font></span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c7">ik</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="E3:121_2"/><i><font color="Green" title="E55">A5</font></i>: 
( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">ik</font> &amp; <font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Maroon" title="c7">ik</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c4">K</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c8">z</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="E5:121_2"/><i><font color="Green" title="E56">A6</font></i>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ik</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:121_2"><i><font color="Green" title="E55">A5</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">i'</font> = <font color="Maroon" title="c7">ik</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font> <b>by </b><i><a class="txt" href="ideal_1.html#E5:121_2"><i><font color="Green" title="E56">A6</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">k'</font> = <font color="Maroon" title="c7">ik</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">K</font> <b>by </b><i><a class="txt" href="ideal_1.html#E5:121_2"><i><font color="Green" title="E56">A6</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">j'</font> = <font color="Maroon" title="c6">j</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font> <b>by </b><i><a class="txt" href="ideal_1.html#E1:121"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="ideal_1.html#E3:121_2"><i><font color="Green" title="E55">A5</font></i></a></i>;<br/><a NAME="E9:121_2"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">i'</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:121_2"><i><font color="Green" title="E55">A5</font></i></a></i>;<br/><a NAME="E10:121_2"/><b>then </b><i><font color="Green" title="E57">A7</font></i>: 
<font color="Maroon" title="c5">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="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">Def1</a></i>;<br/><a NAME="E11:121_2"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">k'</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:121_2"><i><font color="Green" title="E55">A5</font></i></a></i>;<br/><a NAME="E12:121_2"/><b>then </b>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c4">K</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:121_2"><i><font color="Green" title="E55">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E13:121_2"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c4">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="ideal_1.html#E10:121_2"><i><font color="Green" title="E57">A7</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="E4:121"/>
<font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c4">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c4">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="ideal_1.html#E2:121"><i><font color="Green" title="E54">A2</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/>


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