<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">I</font> is  <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a>  iff <font color="Olive" title="b1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">I</font> is  <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a>  iff <font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">P</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>
<a NAME="E1:31"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#D1" title="YELLOW_1:def.1">YELLOW_1:def 1</a></span>;<br/>
<a NAME="E2:31"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">i</font> = <font color="Maroon" title="c2">I</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:31"/>
( <font color="Maroon" title="c2">I</font> is  <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a>  implies <font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> implies <font color="Maroon" title="c2">I</font> is  <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:31_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_7.html#D1" target="_self" title="WAYBEL_7:def.1">WAYBEL_7:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:31_1"/>
<font color="Maroon" title="c4">i</font> is  <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_6.html#D6" title="WAYBEL_6:def.6">WAYBEL_6:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> )</span><br/>

<a NAME="E1:31_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>;<br/>
<a NAME="E2:31_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A3</font></span>: 
 ex <font color="Olive" title="b1">J</font> being  <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">J</font>
 
;<br/>
<a NAME="E3:31_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>;<br/>
<a NAME="E4:31_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">J</font> being  <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">J</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c6">y</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:31_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> )</span><br/>

<a NAME="E7:31_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<a NAME="E8:31_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c7">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T45" title="YELLOW_2:th.45">YELLOW_2:45</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:31_1_1"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
not <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E10:31_1_1"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
not <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E11:31_1_1"/>
not <font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31_1_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:31_1_1"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 <span class="kw">and </span><br/><a NAME="E14:31_1_1"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
not <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E15:31_1_1"/>
not <font color="Maroon" title="c8">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31_1_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E17:31_1_1"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <span class="kw">and </span><br/><a NAME="E18:31_1_1"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
not <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c10">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:31_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E17:31_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a></span>;<br/>
<a NAME="E20:31_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/>
<a NAME="E21:31_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:31_1_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></span>;<br/>
<a NAME="E22:31_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/>
<a NAME="E23:31_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:31_1_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></span>;<br/>
<a NAME="E24:31_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:31_1_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:31_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E8:31_1_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E14:31_1_1"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E18:31_1_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:31_1"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_6.html#D7" title="WAYBEL_6:def.7">WAYBEL_6:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E5:31"/><span class="lab"><font color="Green" title="E16">A12</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">I</font> is  <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_7.html#D1" target="_self" title="WAYBEL_7:def.1">WAYBEL_7:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">X</font> =  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">Y</font> =  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c6">y</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E7:31"/>
( <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c8">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c7">X</font>, <font color="Maroon" title="c10">Y</font> = <font color="Maroon" title="c8">Y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>;<br/>
<a NAME="E9:31"/><span class="lab"><font color="Green" title="E17">A13</font></span>: 
<font color="Maroon" title="c9">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">X</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c10">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T45" title="YELLOW_2:th.45">YELLOW_2:45</a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:31"/><span class="lab"><font color="Green" title="E18">A14</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )</span><br/>

<a NAME="E11:31"/>
<font color="Maroon" title="c9">X</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c10">Y</font> or <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c10">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font></span><br/>

<a NAME="E2:31_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A16</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:31_2"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E1:31_2"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">a</font> = <font color="Maroon" title="c11">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a></span>;<br/>
<a NAME="E5:31_2"/><span class="lab"><font color="Green" title="E22">A18</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_2"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/>
<a NAME="E6:31_2"/>
<font color="Maroon" title="c12">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/>
<a NAME="E7:31_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:31_2"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A19</font></span>: 
<font color="Maroon" title="c9">X</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c10">Y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<a NAME="E13:31"/>
<font color="Maroon" title="c4">i</font> is  <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="waybel_6.html#D7" title="WAYBEL_6:def.7">WAYBEL_6:def 7</a></span>;<br/>
<a NAME="E14:31"/><span class="kw">then </span>
( <font color="Maroon" title="c9">X</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c10">Y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:31"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="ref" href="waybel_6.html#D6" title="WAYBEL_6:def.6">WAYBEL_6:def 6</a></span>;<br/>
<a NAME="E15:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A20</font></span>: 
( <font color="Maroon" title="c9">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<a NAME="E16:31"/>
<font color="Maroon" title="c6">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font>
 
;<br/>
<a NAME="E17:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A21</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/>
<a NAME="E18:31"/>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font>
 
;<br/>
<a NAME="E19:31"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:31"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:31"><span class="lab"><font color="Green" title="E21">A21</font></span></a>, <a class="txt" href="#E15:31"><span class="lab"><font color="Green" title="E20">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
