<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</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>9</sub></font> = the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="cat_5.html#E2:13"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</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#E2:13"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E1:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_2.html#T12">CAT_2:12</a></i>;<br/>
<a NAME="E4:13_1_1"/>
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:13_1_1"/>
(  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U6">Id</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>1</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:13_1_1"/><b>then </b><i><font color="Green">E8</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E8:13_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E1:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_2.html#T14">CAT_2:14</a></i>;<br/>
<a NAME="E7:13_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</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="cat_5.html#E3:13_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:13_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>17</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>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>16</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_2.html#T15">CAT_2:15</a></i>;<br/>
<b>hence </b><a NAME="E9:13_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E10:13_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1_2"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</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="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_2"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E1:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_2.html#T14">CAT_2:14</a></i>;<br/>
<a NAME="E7:13_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:13_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>18</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>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_2.html#T15">CAT_2:15</a></i>;<br/>
<b>hence </b><a NAME="E9:13_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E12:13_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1_3"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_3"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</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#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_3"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E1:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_2.html#T12">CAT_2:12</a></i>;<br/>
<a NAME="E7:13_1_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_3"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:13_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:13_1_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E14:13_1_1"/><i><font color="Green">E9</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</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></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E15:13_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</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></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</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></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E16:13_1_1"/><b>then </b><i><font color="Green">E10</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/>
<a NAME="E17:13_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1_4"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:13_1_1_4"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E16:13_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_4"><i><font color="Green">E11</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K2">`2</a>  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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K2">`2</a>  as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E4:13_1_1_4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E1:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_2.html#T14">CAT_2:14</a></i>;<br/>
<a NAME="E8:13_1_1_4"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E14:13_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_4"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:13_1_1_4"/><b>then </b><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_4"><i><font color="Green">E11</font></i></a>, <a class="txt" href="cat_5.html#E4:13_1_1_4"><i><font color="Green">E12</font></i></a>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E10:13_1_1_4"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>22</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>22</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:13"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cat_2.html#T15">CAT_2:15</a></i>;<br/>
<a NAME="E11:13_1_1_4"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:13"><i><font color="Green">E6</font></i></a>, <a class="txt" href="cat_5.html#E4:13_1_1_4"><i><font color="Green">E12</font></i></a>, <a class="txt" href="cat_5.html#E8:13_1_1_4"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_5.html#E9:13_1_1_4"><i><font color="Green">E14</font></i></a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a>, <a class="ref" href="cat_2.html#T17">CAT_2:17</a></i>;<br/>
<b>hence </b><a NAME="E12:13_1_1_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E16:13_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #);<br/>
<a NAME="E19:13_1_1"/>
 <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) is <a href="cat_1.html#V2">Category-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_5"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #);<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>18</sub></font> 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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E2:13_1_1_5_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1_1_5_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #)
 ;<br/><a NAME="E2:13_1_1_5_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E14:13_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b>the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> = 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E4:13_1_1_5_1"/>
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><a NAME="E5:13_1_1_5_1"/><b>then </b>
( the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E6:13_1_1_5_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</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></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E7:13_1_1_5_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #)
 <b>by </b><i><a class="txt" href="cat_5.html#E14:13_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #);<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>18</sub></font> 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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>assume </b><a NAME="E2:13_1_1_5_2"/><i><font color="Green">E13</font></i>: 
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><a NAME="E3:13_1_1_5_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #)
 <b>by </b><i><a class="txt" href="cat_5.html#E1:13_1_1_5"><i><font color="Green">E11</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <b>by </b><i><a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/><a NAME="E5:13_1_1_5_2"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="cat_2.html#K13">]</a></span> &amp; the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E3:13_1_1_5_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E6:13_1_1_5_2"/>
( the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</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="cat_5.html#E2:13_1_1_5_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #);<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>18</sub></font> 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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>assume </b><a NAME="E2:13_1_1_5_3"/><i><font color="Green">E13</font></i>: 
( the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><a NAME="E3:13_1_1_5_3"/><b>then </b><i><font color="Green">E14</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) )
 <b>by </b><i><a class="txt" href="cat_5.html#E1:13_1_1_5"><i><font color="Green">E11</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">c<sub>25</sub></font> = the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <b>by </b><i><a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>25</sub></font> 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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:13_1_1_5_3"/>
( the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</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="cat_5.html#E2:13_1_1_5"><i><font color="Green">E12</font></i></a>, <a class="txt" href="cat_5.html#E2:13_1_1_5_3"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E7:13_1_1_5_3"/><b>then </b><i><font color="Green">E15</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) )
 <b>by </b><i><a class="txt" href="cat_5.html#E1:13_1_1_5"><i><font color="Green">E11</font></i></a>, <a class="txt" href="cat_5.html#E2:13_1_1_5_3"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E8:13_1_1_5_3"/><i><font color="Green">E16</font></i>: 
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>thus </b>the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1_5_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="cat_2.html#K13">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_5_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="cat_2.html#K13">]</a></span></span>)</span>,<font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E2:13_1_1_5_3"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cat_5.html#E8:13_1_1_5_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E3:13_1_1_5_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1_5_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>thus </b>the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> = 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</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="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
;<br/>

<b>thus </b>the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> = 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</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="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>
;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #);<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> 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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>assume </b><a NAME="E2:13_1_1_5_6"/>
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><a NAME="E3:13_1_1_5_6"/><b>then </b>
( the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:13_1_1_5_6"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</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></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:13_1_1_5_6"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E14:13_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E6:13_1_1_5_6"/>
the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #);<br/>

<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> 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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>assume </b><a NAME="E9:13_1_1_5"/>
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<a NAME="E10:13_1_1_5"/><b>then </b>
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="cat_1.html#D8">CAT_1:def 8</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:13_1_1_5"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</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></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:13_1_1_5"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E7:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="cat_5.html#E14:13_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:13_1_1_5"/>
the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:13_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="cat_1.html#V1">strict</a> <a href="cat_1.html#NM4">Category</a> st <br/>( the <a href="cat_1.html#U1">Objects</a> of <font color="Olive">b<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>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U3">Dom</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U6">Id</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


</div>
