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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="yellow21.html#V4" title="YELLOW21:attr.4">with_all_isomorphisms</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>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="E1:29"/><i><font color="Green" title="E10">A1</font></i>: 
 <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>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">g</font> being  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <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>,<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><b> such that </b><br/><a NAME="E3:29"/><i><font color="Green" title="E11">A2</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> <font color="Maroon" title="c5">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> &amp; <font color="Maroon" title="c5">g</font> <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="funct_2.html#R2" title="FUNCT_2:pred.2">=</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> )
 <b>by </b><i><a class="ref" href="yellow16.html#T17" title="YELLOW16:th.17">YELLOW16:17</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green" title="E12">A3</font></i>: 
(  <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/>
<a NAME="E5:29"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c5">g</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#E3:29"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="yellow16.html#T17" title="YELLOW16:th.17">YELLOW16:17</a></i>;<br/>
<a NAME="E6:29"/><b>then </b><i><font color="Green" title="E14">A5</font></i>: 
(  <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E1:29"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="yellow21.html#D8" target="_self" title="YELLOW21:def.8">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <b>by </b><i><a class="txt" href="yellow21.html#E5:29"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="yellow21.html#D8" target="_self" title="YELLOW21:def.8">Def8</a></i>;<br/>
<a NAME="E8:29"/>
 <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E6:29"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="yellow21.html#D7" target="_self" title="YELLOW21:def.7">Def7</a></i>;<br/>
<a NAME="E9:29"/><b>then </b><i><font color="Green" title="E15">A6</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">g</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="c3">b</font> &amp; <font color="Maroon" title="c6">g</font> <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> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E3:29"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="yellow21.html#E4:29"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="yellow21.html#E6:29"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="yellow21.html#T3" target="_self" title="YELLOW21:th.3">Th3</a></i>;<br/>
<a NAME="E10:29"/><b>then </b><i><font color="Green" title="E16">A7</font></i>: 
( <font color="Maroon" title="c6">g</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">g</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</a></i>;<br/>
<a NAME="E11:29"/><b>then </b>
( <font color="Maroon" title="c4">f</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> &amp; <font color="Maroon" title="c4">f</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> )
 
<b>by </b><i><a class="ref" href="altcat_3.html#D2" title="ALTCAT_3:def.2">ALTCAT_3:def 2</a>, <a class="ref" href="altcat_3.html#D3" title="ALTCAT_3:def.3">ALTCAT_3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:29"/>
( <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> )
 <b>by </b><i><a class="txt" href="yellow21.html#E6:29"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="yellow21.html#E9:29"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="yellow21.html#E10:29"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="altcat_3.html#D4" title="ALTCAT_3:def.4">ALTCAT_3:def 4</a></i>; <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/>


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