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<a NAME="E1:29_1"/><i><font color="Green">E39</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11">ALTCAT_2:def 11</a></i>;<br/>
<a NAME="E2:29_1"/><i><font color="Green">E51</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">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">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow20.html#E1:29"><i><font color="Green">E25</font></i></a>, <a class="ref" href="yellow20.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:29_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>]
 <b>by </b><i/>;<br/><a NAME="E5:29_1_1_1"/><b>then </b>
<font color="Maroon">a</font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:29_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>3</sub></font>()
 ;<br/></div>
<b>end;</b></div>

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

<b>assume </b><a NAME="E2:29_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>3</sub></font>()
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>3</sub></font>() ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow20.html#E1:29"><i><font color="Green">E25</font></i></a>, <a class="ref" href="yellow20.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E5:29_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E6:29_1_1"/><b>then </b>
<font color="Maroon">a</font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:29_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>2</sub></font>(), <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">F<sub>2</sub></font>();<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">a</font>, <font color="Maroon">y1</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/><b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x1</font>, <font color="Maroon">y2</font> = <font color="Maroon">y1</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>3</sub></font>() <b>by </b><i/>;<br/><a NAME="E3:29_1_2"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>() )
 ;<br/><b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow20.html#E1:29"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E5:29_1_2"/><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</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> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</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><a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/><a NAME="E6:29_1_2"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="altcat_1.html#K1">^&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">f</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_2_1"/><i><font color="Green">E56</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">x1</font>,<font color="Maroon">y1</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>

<b>then 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">a'</font>,<font color="Maroon">b'</font> <b>by </b><i><a class="ref" href="yellow20.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:29_1_2_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">f</font>]
 
<b>by </b><i>, <a class="ref" href="yellow20.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>hence </b><a NAME="E4:29_1_2_1"/>
<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">x2</font>,<font color="Maroon">y2</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="yellow20.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E7:29_1_2"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">y2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">y1</font></span><a href="altcat_1.html#K1">^&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">f</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_2_2"/><i><font color="Green">E58</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">x2</font>,<font color="Maroon">y2</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>

<b>then 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">a'</font>,<font color="Maroon">b'</font> <b>by </b><i><a class="ref" href="yellow20.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:29_1_2_2"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">f</font>]
 
<b>by </b><i>, <a class="ref" href="yellow20.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>hence </b><a NAME="E4:29_1_2_2"/>
<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">x1</font>,<font color="Maroon">y1</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="yellow20.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E8:29_1_2"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">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">Arrows</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i>, <a class="ref" href="altcat_1.html#T9">ALTCAT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E5:29_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>by </b><i>, <a class="ref" href="altcat_2.html#T27">ALTCAT_2:27</a></i>;<br/>


</div>
