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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] );<br/>
<div><i><font color="Green">E4</font></i>: <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>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:3_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E2:3_1_1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="uniroots.html#E1:3_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="uniroots.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:3_1_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:3_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/><a NAME="E3:3_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="uniroots.html#E2:3_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="uniroots.html#E1:3_1_1_1"><i><font color="Green">E7</font></i></a></i>;<br/><b>hence </b><a NAME="E4:3_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/></div><b>end;</b></div><a NAME="E5:3_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="uniroots.html#E1:3"><i><font color="Green">E3</font></i></a></i>;<br/><b>hence </b><a NAME="E6:3_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E2:3_1"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S10">NAT_1:sch 10</a>(<a class="txt" href="uniroots.html#E1:3_1"><i><font color="Green">E4</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E3:3_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="uniroots.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E4:3_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:3_1"><i><font color="Green">E5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/>


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