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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:78_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>() <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>1</sub></font>];<br/>
<a NAME="E2:78_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">F<sub>1</sub></font>() as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:78_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>]
 ;<br/><a NAME="E2:78_1_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E3:78_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="nat_1.html#K1" target="_self">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1" target="_self">+</a> 1]
 <b>by </b><i>, <a class="ref" href="nat_1.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E4:78_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font> <a href="nat_1.html#K1" target="_self">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E5:78_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" target="_self">NAT_1:sch 1</a>(, <a class="ref" href="nat_1.html#T18" target="_self">Th18</a>);<br/></i>
<b>consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:78_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="nat_1.html#E4:78_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E8:78_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E9:78_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">i</font>]
 <b>by </b><i><a class="ref" href="nat_1.html#T20" target="_self">Th20</a>, <a class="ref" href="nat_1.html#T22" target="_self">Th22</a></i>;<br/>


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