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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><a NAME="E2:35_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="substlat.html#E2"><i><font color="Green">Lemma1</font></i></a></i>;<br/><a NAME="E3:35_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E1:35_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="substlat.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E4:35_1"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:35_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E29"><i><font color="Green">Lemma24</font></i></a></i>;<br/><a NAME="E5:35_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E3:35_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:35_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E1:35_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="substlat.html#E3:35_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="substlat.html#E4:35_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="substlat.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:35_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:35_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="substlat.html#E1:35_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="substlat.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:35_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E2:35_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="substlat.html#E4:35_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="substlat.html#E5:35_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="substlat.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:35"/>
 <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E11"><i><font color="Green">Lemma8</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:35_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="substlat.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E3:35_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:35_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="substlat.html#E1:35_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="substlat.html#E2"><i><font color="Green">Lemma1</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:35_2_1"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>assume </b><a NAME="E3:35_2_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_2_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:35_2_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E29"><i><font color="Green">Lemma24</font></i></a></i>;<br/><a NAME="E5:35_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E3:35_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="substlat.html#E4:35_2_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:35_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E1:35_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="substlat.html#E3:35_2_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="substlat.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E7:35_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E2:35_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="substlat.html#E3:35_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="substlat.html#E4:35_2_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:35_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="substlat.html#E1:35_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="substlat.html#E2:35_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="substlat.html#E3:35_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="substlat.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:35_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="substlat.html#E3:35_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="substlat.html#E4:35_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="substlat.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:35"/>
 <a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="substlat.html#K3" target="_self">mi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="substlat.html#E11"><i><font color="Green">Lemma8</font></i></a></i>;<br/>


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