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<b>set </b><font color="Maroon">alt</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon">IT</font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon">G</font> = the <a href="altcat_1.html#U1">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">ALTCAT_1:def 9</a>,<a class="ref" href="altcat_1.html#D18">ALTCAT_1:def 18</a>
<div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><b>consider </b><font color="Maroon">f</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:3_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">j'</font>,<font color="Maroon">j'</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>and </b><br/><a NAME="E4:3_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">j'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b</font>,<font color="Maroon">j'</font>,<font color="Maroon">j'</font>,<font color="Olive">g</font>,<font color="Maroon">f</font>) <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">f</font> = <font color="Maroon">f</font>;<br/><b>thus </b><a NAME="E5:3_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">j</font>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E6:3_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/><b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as   <a href="altcat_1.html#NM1">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">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">j'</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/><a NAME="E9:3_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">j</font>,<font color="Maroon">g</font>,<font color="Maroon">f</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">j'</font>,<font color="Maroon">j'</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">i'</font>,<font color="Maroon">j'</font> <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> = 
<font color="Maroon">f'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">g'</font>
<b>by </b><i>, , <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon">g</font>
<b>by </b><i>, , , </i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">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">ALTCAT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:3_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">i'</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>and </b><br/><a NAME="E5:3_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">i'</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <br/><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">i</font>,<font color="Olive">b</font>,<font color="Maroon">e</font>,<font color="Olive">g</font>) <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<b>thus </b><a NAME="E6:3_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">i</font>
 <b>by </b><i/>;<br/>

<b>reconsider </b><font color="Maroon">e'</font> = <font color="Maroon">e</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">i'</font>,<font color="Maroon">i'</font> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E8:3_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E10:3_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">j'</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
;<br/>
<a NAME="E11:3_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">e</font>,<font color="Maroon">f</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">i'</font>,<font color="Maroon">j'</font> <b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">e</font> = 
<font color="Maroon">f'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">e'</font>
<b>by </b><i>, , <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i>, , , </i>
;<br/>


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