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<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> ] means verum;<br/>
<a NAME="E1:31_1"/><i><font color="Green">E23</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="yellow20.html#E1:31"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E2:31_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>] &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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>  holds <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>4</sub></font>]
 
;<br/>
<a NAME="E3:31_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>, <a href="altcat_1.html#K8">idm</a> <font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V6">strict</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:31_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/>( <font color="Olive">b<sub>1</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 <b>and </b><br/><a NAME="E6:31_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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>  holds <br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>5</sub></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<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font>] )
 <b>from </b><i><a class="ref" href="yellow18.html#S7">YELLOW18:sch 7</a>(<a class="txt" href="yellow20.html#E1:31_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="yellow20.html#E2:31_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow20.html#E3:31_1"><i><font color="Green">E25</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:31_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E2:31_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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="E3:31_1_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></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="E1:31_1_1_1"/>
<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:31_1_1_1"/>
<font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow20.html#E6:31_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="yellow20.html#E1:31_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:31_1_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow20.html#E2:31_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:31_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is <a href="altcat_2.html#V2">full</a>
 
<b>by </b><i><a class="ref" href="yellow20.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E9:31_1"/>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V6">strict</a> <a href="altcat_2.html#V2">full</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/>( <font color="Olive">b<sub>2</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Olive">b<sub>1</sub></font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] )
 <b>by </b><i><a class="txt" href="yellow20.html#E5:31_1"><i><font color="Green">E26</font></i></a></i>;<br/>


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