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<a NAME="E1:1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">ff</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">t</font>,<font color="Olive">ff</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>proof </b></a><div class="add">

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

<b>consider </b><font color="Maroon">ff</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E2:1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Maroon">ff</font>]
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">ff</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E4:1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Maroon">ff</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:1_1_1"/>
 ex <font color="Olive">ff</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Olive">ff</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">t</font>,<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">t</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

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

<b>thus </b><a NAME="E4:1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>]
 <b>by </b><i/>;<br/>


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