<?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_2.html#V1">reflexive</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#V1">reflexive</a>  <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:8"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</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>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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E2:8"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a></i>;<br/>
<a NAME="E3:8"/><b>then </b><i><font color="Green">E4</font></i>: 
<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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</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="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<i><font color="Green">E5</font></i>: the <a href="functor0.html#U1">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="partfun1.html#K6">id</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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="functor1.html#E1:8"><i><font color="Green">E3</font></i></a>, <a class="ref" href="functor0.html#D29">FUNCTOR0:def 29</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><span class="p2"><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> <a href="xboole_0.html#K3">/\</a> <span class="p2"><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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E3:8"><i><font color="Green">E4</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2"><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>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2"><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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T43">FUNCT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2"><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>)</span> <a href="partfun1.html#K1">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#D29">FUNCTOR0:def 29</a></i>
<br/>.= 
the <a href="functor0.html#U1">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#D29">FUNCTOR0:def 29</a></i>
;<br/>
<a NAME="E5:8"/>
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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/>the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="altcat_2.html#K3">**</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</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="8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>7</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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:8_2"/><b>then </b><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2"><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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E3:8_2"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font> <a href="altcat_2.html#R2">cc=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a></i>;<br/>
<a NAME="E4:8_2"/><b>then </b><i><font color="Green">E8</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</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> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="functor1.html#E1:8_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="altcat_2.html#D2">ALTCAT_2:def 2</a></i>;<br/>
<a NAME="E5:8_2"/><i><font color="Green">E9</font></i>: 
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:8_2_1"/><i><font color="Green">E10</font></i>: 
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="functor0.html#D29">FUNCTOR0:def 29</a></i>;<br/>
<a NAME="E2:8_2_1"/><i><font color="Green">E11</font></i>: 
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="functor0.html#D29">FUNCTOR0:def 29</a></i>;<br/>
<a NAME="E3:8_2_1"/><i><font color="Green">E12</font></i>: 
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="functor0.html#D29">FUNCTOR0:def 29</a></i>;<br/>
<span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E1:8_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="functor1.html#E1:8_2_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="functor1.html#E2:8_2_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default">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> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E3:8"><i><font color="Green">E4</font></i></a>, <a class="txt" href="functor1.html#E1:8_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">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> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T43">FUNCT_1:43</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E4:8_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="functor1.html#E1:8"><i><font color="Green">E3</font></i></a>, <a class="txt" href="functor1.html#E1:8_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="functor1.html#E3:8_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="msualg_3.html#D1">MSUALG_3:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E5:8_2_1"/>
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<i><font color="Green">E10</font></i>: <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</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>7</sub></font> = 
<span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3"><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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="functor0.html#D29">FUNCTOR0:def 29</a></i>
<br/>.= 
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3"><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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor1.html#E2:8_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="functor1.html#E1:8_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E7:8_2"/><i><font color="Green">E11</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="altcat_2.html#K3">**</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>;<br/>
<a NAME="E8:8_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:8_2_3"/><i><font color="Green">E12</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3"><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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#D29">FUNCTOR0:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T37">FUNCT_1:37</a></i>
<br/>.= 
<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> <a href="xboole_0.html#K3">/\</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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<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>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="functor1.html#E3:8"><i><font color="Green">E4</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E3:8_2_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="functor1.html#E1:8_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="functor1.html#E1:8_2_3"><i><font color="Green">E12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:8_2"/>
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p3">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="altcat_2.html#K3">**</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</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>7</sub></font>
 <b>by </b><i><a class="txt" href="functor1.html#E5:8_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="functor1.html#E6:8_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="functor1.html#E7:8_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:8"/><b>then </b>
the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="functor0.html#U2">MorphMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#K7">*</a> the <a href="functor0.html#U1">ObjectMap</a> of <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="altcat_2.html#K3">**</a> the <a href="functor0.html#U2">MorphMap</a> of <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="functor1.html#E1:8"><i><font color="Green">E3</font></i></a>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>hence </b><a NAME="E7:8"/>
 <a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K13">*</a> <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="functor1.html#E1:8"><i><font color="Green">E3</font></i></a>, <a class="txt" href="functor1.html#E4:8"><i><font color="Green">E5</font></i></a>, <a class="ref" href="functor0.html#D37">FUNCTOR0:def 37</a></i>;<br/>


</div>
