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<b>let </b><font color="Maroon">C</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">D</font> be    <a href="altcat_2.html#M1" target="_self">SubCatStr</a> of <font color="Maroon">C</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green">E16</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green">E17</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font>
 ;<br/>

<a NAME="E3:62"/><i><font color="Green">E20</font></i>: 
not <font color="Maroon">D</font> is <a href="struct_0.html#V3">empty</a>
 
<b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E4:62"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">D</font> is <a href="altcat_1.html#V2">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">D</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">D</font>, <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a><br/>

<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">o1</font>, <font color="Maroon">p2</font> = <font color="Maroon">o2</font>, <font color="Maroon">p3</font> = <font color="Maroon">o3</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> <b>by </b><i><a class="txt" href="altcat_2.html#E3:62_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E2:62_1"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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">p1</font>,<font color="Maroon">p2</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:62_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E3:62_1"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</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">p2</font>,<font color="Maroon">p3</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:62_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E4:62_1"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</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">p1</font>,<font color="Maroon">p3</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:62_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>assume </b><a NAME="E5:62_1"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</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>  )
 ;<br/>

<b>hence </b><a NAME="E6:62_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</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> 
 <b>by </b><i>, , , <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:62"/>
<font color="Maroon">C</font> is    <a href="altcat_2.html#M1" target="_self">SubCatStr</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:62"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E6:62"/><b>then </b>
<font color="Maroon">C</font> is    <a href="altcat_2.html#M1" target="_self">SubCatStr</a> of <font color="Maroon">D</font>
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E7:62"/>
 <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">D</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">D</font>,the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">D</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">C</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font>,the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> #)
 <b>by </b><i/>;<br/>


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