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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow21.html#V2" target="_self">lattice-wise</a> <a href="yellow21.html#V4" target="_self">with_all_isomorphisms</a> <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>





<b>assume </b><a NAME="E1:29"/><i><font color="Green">E10</font></i>: 
 <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:29"/><i><font color="Green">E11</font></i>: 
( <span class="p1">(<span class="default"><a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="yellow16.html#T17">YELLOW16:17</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green">E12</font></i>: 
(  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow21.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E5:29"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 
<b>by </b><i><a class="txt" href="yellow21.html#E3:29"><i><font color="Green">E11</font></i></a>, <a class="ref" href="yellow16.html#T17">YELLOW16:17</a></i>;<br/>
<a NAME="E6:29"/><b>then </b><i><font color="Green">E14</font></i>: 
(  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E1:29"><i><font color="Green">E10</font></i></a>, <a class="ref" href="yellow21.html#D8" target="_self">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="yellow21.html#E5:29"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow21.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E8:29"/>
 <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E6:29"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow21.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E9:29"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E3:29"><i><font color="Green">E11</font></i></a>, <a class="txt" href="yellow21.html#E4:29"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow21.html#E6:29"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow21.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E10:29"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a></i>;<br/>
<a NAME="E11:29"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V1">retraction</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="altcat_3.html#V2">coretraction</a> )
 
<b>by </b><i><a class="ref" href="altcat_3.html#D2">ALTCAT_3:def 2</a>, <a class="ref" href="altcat_3.html#D3">ALTCAT_3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:29"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="yellow21.html#E6:29"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow21.html#E9:29"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow21.html#E10:29"><i><font color="Green">E16</font></i></a>, <a class="ref" href="altcat_3.html#D4">ALTCAT_3:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D5">ALTCAT_3:def 5</a><br/>


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