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<b>let </b><font color="Maroon" title="c2">B</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">A</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">B</font> is <a href="yellow21.html#V2" title="YELLOW21:attr.2">lattice-wise</a></span><br/>

<b>thus </b><a NAME="E1:58_1_1"/>
( <font color="Maroon" title="c2">B</font> is <a href="altcat_1.html#V9" title="ALTCAT_1:attr.9">semi-functional</a> &amp; <font color="Maroon" title="c2">B</font> is <a href="yellow18.html#V3" title="YELLOW18:attr.3">set-id-inheriting</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow21.html#D4" target="_self" title="YELLOW21:def.4">YELLOW21:def 4</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">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font> holds  <font color="Olive" title="b1">a</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>  st <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font> ) )</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"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>  st <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font></span>
<div class="add"><b>let </b><font color="Maroon" title="c3">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</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">b</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a></span><br/><b>reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="altcat_2.html#T30" title="ALTCAT_2:th.30">ALTCAT_2:30</a></i>;<br/><a NAME="E2:58_1_1_1"/>
<font color="Maroon" title="c4">a</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 <b>by </b><i><a class="ref" href="yellow21.html#D4" target="_self" title="YELLOW21:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_1"/>
<font color="Maroon" title="c3">b</font> is   <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"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</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">A</font>, <font color="Olive" title="b2">B</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> holds <br/><span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">b'</font> = <font color="Maroon" title="c4">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="altcat_2.html#T30" title="ALTCAT_2:th.30">ALTCAT_2:30</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be   <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"> ( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> )</span><br/>

<b>assume </b><a NAME="E4:58_1_1"/>
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><br/>

<a NAME="E5:58_1_1"/><b>then </b>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> )
 
<b>by </b><i><a class="ref" href="yellow21.html#D4" target="_self" title="YELLOW21:def.4">Def4</a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></i>;<br/>
<b>hence </b><a NAME="E6:58_1_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="orders_3.html#K1" title="ORDERS_3:func.1">MonFuncs</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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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