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<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:3_1"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E1:3"><i><font color="Green">E1</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ;<br/>
<a NAME="E3:3_1"/><i><font color="Green">E3</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>from </b><i><a class="ref" href="asympt_0.html#S2" target="_self">ASYMPT_0:sch 2</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:3_1_1_1"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>1</sub></font>() )
 ;<br/><a NAME="E4:3_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:3_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="asympt_0.html#E3:3_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E5:3_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:3_1_1_1"><i><font color="Green">E4</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:3_1_1"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E5:3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="asympt_0.html#E4:3_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E6:3_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> 
 <b>by </b><i><a class="txt" href="asympt_0.html#E2:3_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="asympt_0.html#E4:3_1_1"><i><font color="Green">E4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:3_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="asympt_0.html#E3:3_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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