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

<b>set </b><font color="Maroon" title="c1">alt</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c2">IT</font> = the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c3">I</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c4">G</font> = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D9" title="ALTCAT_1:def.9">ALTCAT_1:def 9</a>,<a class="ref" href="altcat_1.html#D18" title="ALTCAT_1:def.18">ALTCAT_1:def 18</a>
<div class="add"><b>let </b><font color="Maroon" title="c5">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>reconsider </b><font color="Maroon" title="c6">j'</font> = <font color="Maroon" title="c5">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><b>consider </b><font color="Maroon" title="c7">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:3_1_1"/><i><font color="Green" title="E4">A4</font></i>: 
<font color="Maroon" title="c7">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="c6">j'</font>,<font color="Maroon" title="c6">j'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>and </b><br/><a NAME="E4:3_1_1"/><i><font color="Green" title="E5">A5</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">g</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="Olive" title="b1">b</font>,<font color="Maroon" title="c6">j'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c6">j'</font>,<font color="Maroon" title="c6">j'</font>,<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E3:3"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">f</font>;<br/><b>thus </b><a NAME="E5:3_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E3:3_1_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c10">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E6:3_1_1"/><i><font color="Green" title="E6">A6</font></i>: 
<font color="Maroon" title="c10">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">i'</font> = <font color="Maroon" title="c9">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><a NAME="E8:3_1_1"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">i'</font>,<font color="Maroon" title="c6">j'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><a NAME="E9:3_1_1"/><b>then </b><i><font color="Green" title="E7">A7</font></i>: 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c8">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E4:3_1_1"><i><font color="Green" title="E5">A5</font></i></a>, <a class="txt" href="yellow18.html#E6:3_1_1"><i><font color="Green" title="E6">A6</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">f'</font> = <font color="Maroon" title="c8">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">j'</font>,<font color="Maroon" title="c6">j'</font> <b>by </b><i><a class="txt" href="yellow18.html#E3:3_1_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">g'</font> = <font color="Maroon" title="c10">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">i'</font>,<font color="Maroon" title="c6">j'</font> <b>by </b><i><a class="txt" href="yellow18.html#E6:3_1_1"><i><font color="Green" title="E6">A6</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c10">g</font> = 
<font color="Maroon" title="c12">f'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c13">g'</font>
<b>by </b><i><a class="txt" href="yellow18.html#E3:3_1_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="txt" href="yellow18.html#E6:3_1_1"><i><font color="Green" title="E6">A6</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c10">g</font>
<b>by </b><i><a class="txt" href="yellow18.html#E1:3"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="yellow18.html#E3:3_1_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="txt" href="yellow18.html#E6:3_1_1"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="yellow18.html#E9:3_1_1"><i><font color="Green" title="E7">A7</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(); <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D8" title="ALTCAT_1:def.8">ALTCAT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">i'</font> = <font color="Maroon" title="c5">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>consider </b><font color="Maroon" title="c7">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:3_1"/><i><font color="Green" title="E4">A8</font></i>: 
<font color="Maroon" title="c7">e</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="c6">i'</font>,<font color="Maroon" title="c6">i'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>and </b><br/><a NAME="E5:3_1"/><i><font color="Green" title="E5">A9</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">g</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="c6">i'</font>,<font color="Olive" title="b1">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> holds <br/><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c7">e</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E2:3"><i><font color="Green" title="E2">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">e</font>
;<br/>

<b>thus </b><a NAME="E6:3_1"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E4:3_1"><i><font color="Green" title="E4">A8</font></i></a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c8">e'</font> = <font color="Maroon" title="c7">e</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">i'</font>,<font color="Maroon" title="c6">i'</font> <b>by </b><i><a class="txt" href="yellow18.html#E4:3_1"><i><font color="Green" title="E4">A8</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c9">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c10">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E8:3_1"/><i><font color="Green" title="E6">A10</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">j'</font> = <font color="Maroon" title="c9">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E10:3_1"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">i'</font>,<font color="Maroon" title="c11">j'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
;<br/>
<a NAME="E11:3_1"/><b>then </b><i><font color="Green" title="E7">A11</font></i>: 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c10">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E5:3_1"><i><font color="Green" title="E5">A9</font></i></a>, <a class="txt" href="yellow18.html#E8:3_1"><i><font color="Green" title="E6">A10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">f'</font> = <font color="Maroon" title="c10">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">i'</font>,<font color="Maroon" title="c11">j'</font> <b>by </b><i><a class="txt" href="yellow18.html#E8:3_1"><i><font color="Green" title="E6">A10</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c7">e</font> = 
<font color="Maroon" title="c12">f'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">e'</font>
<b>by </b><i><a class="txt" href="yellow18.html#E4:3_1"><i><font color="Green" title="E4">A8</font></i></a>, <a class="txt" href="yellow18.html#E8:3_1"><i><font color="Green" title="E6">A10</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c10">f</font>
<b>by </b><i><a class="txt" href="yellow18.html#E1:3"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="yellow18.html#E4:3_1"><i><font color="Green" title="E4">A8</font></i></a>, <a class="txt" href="yellow18.html#E8:3_1"><i><font color="Green" title="E6">A10</font></i></a>, <a class="txt" href="yellow18.html#E11:3_1"><i><font color="Green" title="E7">A11</font></i></a></i>
;<br/>


</div>
