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<a NAME="E1:32_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Olive">a</font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">a</font>] )
 
<b>by </b><i/>;<br/>
<a NAME="E2:32_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Olive">a</font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>3</sub></font>() iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">a</font>] )
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">a'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon">b'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>();<br/><b>assume </b><a NAME="E1:32_1_1"/>
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/><a NAME="E2:32_1_1"/><b>then </b>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:32_1_1"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> holds <br/> ( <font color="Olive">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">a'</font>,<font color="Maroon">b'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">f</font>] ) )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">a'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>3</sub></font>(), <font color="Maroon">b'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>3</sub></font>();<br/><b>assume </b><a NAME="E1:32_1_2"/>
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/><a NAME="E2:32_1_2"/><b>then </b>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:32_1_2"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> holds <br/> ( <font color="Olive">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">a'</font>,<font color="Maroon">b'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">f</font>] ) )
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:32_1"/>
 <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>(),the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">F<sub>2</sub></font>() #) <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>3</sub></font>(),the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>3</sub></font>(),the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">F<sub>3</sub></font>() #)
 <b>from </b><i><a class="ref" href="yellow20.html#S1" target="_self">YELLOW20:sch 1</a>(, , <a class="txt" href="yellow20.html#E1:32"><i><font color="Green">E25</font></i></a>, <a class="ref" href="yellow20.html#T2" target="_self">Th2</a>);</i><br/>


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