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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="altcat_1.html#L1">AltGraph</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#V1">reflexive</a>  <a href="altcat_1.html#L1">AltGraph</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="functor0.html#V8">feasible</a>  <a href="functor0.html#L2">FunctorStr</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="functor0.html#L2">FunctorStr</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:13"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="functor0.html#V18">full</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="functor0.html#V18">full</a> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:13"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="pboole.html#R6">=</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="msualg_3.html#V2">"onto"</a> )
 <b>by </b><i><a class="txt" href="functor1.html#E1:13"><i><font color="Green">E8</font></i></a>, <a class="ref" href="functor0.html#D33">FUNCTOR0:def 33</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E5:13"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="pboole.html#R6">=</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="msualg_3.html#V2">"onto"</a> )
 <b>by </b><i><a class="txt" href="functor1.html#E1:13"><i><font color="Green">E8</font></i></a>, <a class="ref" href="functor0.html#D33">FUNCTOR0:def 33</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>4</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>5</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E8:13"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="msualg_3.html#V2">"onto"</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="functor0.html#D5">FUNCTOR0:def 5</a></i>;<br/>
<a NAME="E2:13_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="altcat_2.html#K3">**</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="functor1.html#E3:13"><i><font color="Green">E9</font></i></a>, <a class="txt" href="functor1.html#E5:13"><i><font color="Green">E10</font></i></a>, <a class="ref" href="functor0.html#D37">FUNCTOR0:def 37</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<a NAME="E3:13_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><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> ;<br/>

<b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:13_1_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</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="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E3:13_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="altcat_2.html#K3">**</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="functor1.html#E1:13_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <b>by </b><i><a class="txt" href="functor1.html#E2:13_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E5:13_1_1"/><i><font color="Green">E15</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_1_1_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:13_1_1_1_1_1"/><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:13_1_1_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E3:13_1_1_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="functor1.html#E1:13_1_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:13_1_1_1_1_2"/><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:13_1_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="functor1.html#E5:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="functor1.html#E2:13_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/><a NAME="E3:13_1_1_1_1_2"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="functor1.html#E1:13_1_1_1_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:13_1_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_1_2_1"><b>proof </b></a><div class="add">

 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>11</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="txt" href="functor1.html#E1:13_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="functor1.html#E3:13_1_1_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E3:13"><i><font color="Green">E9</font></i></a>, <a class="txt" href="functor1.html#E1:13_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E2:13_1_1_1_1_2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:13_1_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> = 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="pboole.html#K7">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E2:13_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="functor1.html#E3:13_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E1:13_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="functor1.html#E5:13_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E5:13"><i><font color="Green">E10</font></i></a>, <a class="txt" href="functor1.html#E2:13_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>
<br/>.= 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E2:13_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E1:13_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="functor0.html#U1">ObjectMap</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#D37">FUNCTOR0:def 37</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="functor1.html#E1:13_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>
<b>hence </b><a NAME="E7:13_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:13_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="pboole.html#R6">=</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="msualg_3.html#V2">"onto"</a> )
 <b>by </b><i><a class="ref" href="msualg_3.html#D3">MSUALG_3:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:13"/>
<font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K13">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="functor0.html#V18">full</a>
 <b>by </b><i><a class="ref" href="functor0.html#D33">FUNCTOR0:def 33</a></i>;<br/>


</div>
