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

<a NAME="E1:29_1"/><i><font color="Green" title="E25">A5</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11" title="ALTCAT_2:def.11">ALTCAT_2:def 11</a></i>;<br/>
<a NAME="E2:29_1"/><i><font color="Green" title="E26">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_1"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon" title="c2">a</font> = <font color="Maroon" title="c1">x</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/><b>reconsider </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">a</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow20.html#E1:29_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="yellow20.html#E1:29_1_1_1"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/><a NAME="E4:29_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>]
 <b>by </b><i><a class="txt" href="yellow20.html#E1:29"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/><a NAME="E5:29_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="txt" href="yellow20.html#E3:29"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E6:29_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>3</sub></font>()
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:29_1_1"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>3</sub></font>()
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">a</font> = <font color="Maroon" title="c1">x</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>3</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">a</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow20.html#E1:29_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="yellow20.html#E2:29_1_1"><i><font color="Green" title="E27">A8</font></i></a></i>;<br/>
<a NAME="E5:29_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>]
 
<b>by </b><i><a class="txt" href="yellow20.html#E3:29"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/>
<a NAME="E6:29_1_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="txt" href="yellow20.html#E1:29"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:29_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>();<br/><b>reconsider </b><font color="Maroon" title="c3">x1</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c4">y1</font> = <font color="Maroon" title="c2">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/><b>reconsider </b><font color="Maroon" title="c5">x2</font> = <font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c6">y2</font> = <font color="Maroon" title="c4">y1</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>3</sub></font>() <b>by </b><i><a class="txt" href="yellow20.html#E2:29_1"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/><a NAME="E3:29_1_2"/>
( <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon" title="c4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c2">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow20.html#E1:29_1"><i><font color="Green" title="E25">A5</font></i></a></i>;<br/><a NAME="E5:29_1_2"/><i><font color="Green" title="E27">A9</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <b>by </b><i><a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></i>;<br/><a NAME="E6:29_1_2"/><i><font color="Green" title="E28">A10</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_2_1"/><i><font color="Green" title="E29">A11</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font> <b>by </b><i><a class="txt" href="yellow20.html#E5:29_1_2"><i><font color="Green" title="E27">A9</font></i></a></i>;<br/>
<a NAME="E3:29_1_2_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c10">f</font>]
 
<b>by </b><i><a class="txt" href="yellow20.html#E2:29"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="yellow20.html#E5:29_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="yellow20.html#E1:29_1_2_1"><i><font color="Green" title="E29">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_2_1"/>
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow20.html#E4:29"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="yellow20.html#E5:29_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="yellow20.html#E1:29_1_2_1"><i><font color="Green" title="E29">A11</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:29_1_2"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_2_2"/><i><font color="Green" title="E29">A12</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font> <b>by </b><i><a class="txt" href="yellow20.html#E5:29_1_2"><i><font color="Green" title="E27">A9</font></i></a></i>;<br/>
<a NAME="E3:29_1_2_2"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c10">f</font>]
 
<b>by </b><i><a class="txt" href="yellow20.html#E4:29"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="yellow20.html#E5:29_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="yellow20.html#E1:29_1_2_2"><i><font color="Green" title="E29">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_2_2"/>
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">y1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow20.html#E2:29"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="yellow20.html#E5:29_1_2"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="yellow20.html#E1:29_1_2_2"><i><font color="Green" title="E29">A12</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:29_1_2"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="yellow20.html#E6:29_1_2"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:29_1"/><b>then </b>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="txt" href="yellow20.html#E2:29_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="altcat_1.html#T9" title="ALTCAT_1:th.9">ALTCAT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E5:29_1"/>
 <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>2</sub></font>() #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>3</sub></font>(),the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>3</sub></font>(),the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>3</sub></font>() #)
 <b>by </b><i><a class="txt" href="yellow20.html#E2:29_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="altcat_2.html#T27" title="ALTCAT_2:th.27">ALTCAT_2:27</a></i>;<br/>


</div>
