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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="instalg1.html#K1">|</a> <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E1:80"/><i><font color="Green">E44</font></i>: 
 <a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font>, <a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font> <a href="pua2mss1.html#R3">form_morphism_between</a>  <a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="catalg_1.html#T33" target="_self">Th33</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="catalg_1.html#K1" target="_self">-MSF</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="instalg1.html#K1">|</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="catalg_1.html#T38" target="_self">Th38</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E3:80"/>
<font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="catalg_1.html#K1" target="_self">-MSF</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D9">MSUALG_3:def 9</a><br/>

<b>assume </b><a NAME="E4:80"/><i><font color="Green">E45</font></i>: 
 <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="ref" href="catalg_1.html#T25" target="_self">Th25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1"><b>suppose </b></a><a NAME="E1:80_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:80_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E4:80_1_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K8" target="_self">idsym</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:80_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="catalg_1.html#T31" target="_self">Th31</a></i>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><a NAME="E5:80_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp;  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp;  <a href="msualg_1.html#K3">Args</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="instalg1.html#K1">|</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E1:80"><i><font color="Green">E44</font></i></a>, <a class="txt" href="catalg_1.html#E4:80"><i><font color="Green">E45</font></i></a>, <a class="txt" href="catalg_1.html#E3:80_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="catalg_1.html#E4:80_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="catalg_1.html#T34" target="_self">Th34</a>, <a class="ref" href="instalg1.html#T25">INSTALG1:25</a></i>;<br/><a NAME="E6:80_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K5">#</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:80_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>13</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>13</sub></font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="catalg_1.html#E3:80_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="catalg_1.html#E6:80_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="catalg_1.html#D15" target="_self">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="catalg_1.html#E3:80_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="catalg_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="catalg_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="catalg_1.html#D15" target="_self">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K16">hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#D25">CAT_1:def 25</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="catalg_1.html#E7:80_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cat_1.html#T131">CAT_1:131</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K5">#</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="catalg_1.html#E4:80_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="catalg_1.html#E6:80_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="catalg_1.html#D15" target="_self">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="instalg1.html#K1">|</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K5">#</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="catalg_1.html#E1:80"><i><font color="Green">E44</font></i></a>, <a class="ref" href="instalg1.html#T24">INSTALG1:24</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2"><b>suppose </b></a><a NAME="E1:80_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:80_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="catalg_1.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E4:80_1_2"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="catalg_1.html#K10" target="_self">compsym</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="catalg_1.html#E3:80_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="catalg_1.html#T32" target="_self">Th32</a></i>;<br/><b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K3">Args</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:80_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E4:80"><i><font color="Green">E45</font></i></a>, <a class="txt" href="catalg_1.html#E3:80_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="catalg_1.html#T39" target="_self">Th39</a></i>;<br/><a NAME="E7:80_1_2"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E6:80_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><a NAME="E8:80_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E6:80_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>18</sub></font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/><a NAME="E10:80_1_2"/><i><font color="Green">E50</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="catalg_1.html#E6:80_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="catalg_1.html#E3:80_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="catalg_1.html#E6:80_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="catalg_1.html#D15" target="_self">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p2">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="catalg_1.html#E3:80_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="catalg_1.html#T29" target="_self">Th29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K9" target="_self">homsym</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="catalg_1.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="catalg_1.html#D15" target="_self">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K16">hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#D25">CAT_1:def 25</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="catalg_1.html#E8:80_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cat_1.html#T131">CAT_1:131</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="catalg_1.html#E6:80_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i><a class="txt" href="catalg_1.html#E4:80_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="catalg_1.html#E10:80_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="catalg_1.html#D15" target="_self">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K5">#</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="catalg_1.html#E3:80_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="catalg_1.html#E6:80_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="catalg_1.html#E8:80_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="catalg_1.html#T40" target="_self">Th40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="catalg_1.html#K13" target="_self">MSAlg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="instalg1.html#K1">|</a> <span class="p3">(<span class="default"><a href="catalg_1.html#K3" target="_self">CatSign</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="catalg_1.html#K11" target="_self">Upsilon</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="catalg_1.html#K12" target="_self">Psi</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#K5">#</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="catalg_1.html#E1:80"><i><font color="Green">E44</font></i></a>, <a class="ref" href="instalg1.html#T24">INSTALG1:24</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
