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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">l1</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="c2">l1</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> <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font> )</span><br/><b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c2">l1</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> <font color="Maroon" title="c1">L</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">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font></span><br/><b>then reconsider </b><font color="Maroon" title="c3">l</font> = <font color="Maroon" title="c2">l1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font> ;<br/><b>reconsider </b><font color="Maroon" title="c4">l</font> = <font color="Maroon" title="c3">l</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="#E1:47_1"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/><a NAME="E4:47_1"/>
<font color="Maroon" title="c4">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a>
 <b>by </b><i><a class="txt" href="#E1:47_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="waybel_6.html#D7" target="_self" title="WAYBEL_6:def.7">Def7</a></i>;<br/><a NAME="E5:47_1"/><b>then </b>
<font color="Maroon" title="c4">l</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#T27" target="_self" title="WAYBEL_6:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E6:47_1"/>
<font color="Maroon" title="c2">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="waybel_6.html#D4" target="_self" title="WAYBEL_6:def.4">Def4</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="E2:47"/>
 <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">l1</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="c2">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">l1</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> <font color="Maroon" title="c1">L</font> )</span><br/><b>assume </b><a NAME="E1:47_2"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c2">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</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">l1</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> <font color="Maroon" title="c1">L</font></span><br/><b>then reconsider </b><font color="Maroon" title="c3">l</font> = <font color="Maroon" title="c2">l1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font> ;<br/><b>reconsider </b><font color="Maroon" title="c4">l</font> = <font color="Maroon" title="c3">l</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="#E1:47_2"><i><font color="Green" title="E30">A2</font></i></a></i>;<br/><a NAME="E4:47_2"/>
<font color="Maroon" title="c4">l</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a>
 <b>by </b><i><a class="txt" href="#E1:47_2"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="waybel_6.html#D4" target="_self" title="WAYBEL_6:def.4">Def4</a></i>;<br/><a NAME="E5:47_2"/><b>then </b>
<font color="Maroon" title="c4">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#T27" target="_self" title="WAYBEL_6:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E6:47_2"/>
<font color="Maroon" title="c2">l1</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> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="waybel_6.html#D7" target="_self" title="WAYBEL_6:def.7">Def7</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="E4:47"/>
 <a href="waybel_6.html#K3" title="WAYBEL_6:func.3">IRR</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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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