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<b>let </b><font color="Maroon" title="c1">C</font> be  <a href="yellow21.html#V2" title="YELLOW21:attr.2">lattice-wise</a> <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <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="c1">C</font>;<br/>



<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E10">A1</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>

<b>assume </b><a NAME="E2:30"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D5" title="ALTCAT_3:def.5">ALTCAT_3:def 5</a><br/>

<a NAME="E3:30"/><i><font color="Green" title="E12">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E1:30"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="yellow21.html#T3" target="_self" title="YELLOW21:th.3">Th3</a></i>;<br/>
<a NAME="E4:30"/>
(  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c2">a</font></span>)</span> &amp;  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c3">b</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow21.html#T2" target="_self" title="YELLOW21:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E5:30"/>
 <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c4">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="yellow21.html#E2:30"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="yellow21.html#E3:30"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="yellow16.html#T17" title="YELLOW16:th.17">YELLOW16:17</a></i>;<br/>


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