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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="cat_5.html#V3" target="_self">Categorial</a> <a href="cat_5.html#V4" target="_self">full</a> <a href="cat_1.html#NM4">Category</a>;<br/>

<b>assume </b><a NAME="E1:42"/><i><font color="Green">E26</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="cat_5.html#V2" target="_self">categorial</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="cat_5.html#D6" target="_self">Def6</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="cat_5.html#K11" target="_self">`2</a> </span>)</span> where B is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E4:42"/>
<font color="Maroon">c<sub>5</sub></font> <a href="cat_5.html#K11" target="_self">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="cat_5.html#K11" target="_self">`2</a> </span>)</span> where B is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="cat_5.html#K11" target="_self">`2</a> </span>)</span> where B is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> #), <font color="Maroon">c<sub>8</sub></font> =  <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>2</sub></font> #) as  <a href="cat_1.html#V1">strict</a> <a href="cat_1.html#NM4">Category</a> <b>by </b><i><a class="ref" href="cat_5.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:42"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="cat_5.html#V3" target="_self">Categorial</a> &amp; <font color="Maroon">c<sub>8</sub></font> is <a href="cat_5.html#V3" target="_self">Categorial</a> )
 
<b>by </b><i><a class="ref" href="cat_5.html#T10" target="_self">Th10</a></i>;<br/>
<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="cat_5.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E2:42_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="cat_5.html#K11" target="_self">`2</a>  <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E3:42_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> #) iff <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E1:42"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E2:42_2"/>
(  <a href="cat_5.html#K10" target="_self">cat</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="cat_5.html#K10" target="_self">cat</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="cat_5.html#T16" target="_self">Th16</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="cat_5.html#D8" target="_self">Def8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="cat_5.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E5:42_2"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="cat_5.html#K11" target="_self">`2</a>  <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E6:42_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>2</sub></font> #) iff <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:42"/>
 <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>2</sub></font> #)
 <b>by </b><i><a class="txt" href="cat_5.html#E1:42"><i><font color="Green">E26</font></i></a>, <a class="txt" href="cat_5.html#E7:42"><i><font color="Green">E27</font></i></a>, <a class="txt" href="cat_5.html#E8:42"><i><font color="Green">E28</font></i></a>, <a class="ref" href="cat_5.html#T18" target="_self">Th18</a></i>;<br/>


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