<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">D</font>;<br/><b>let </b><font color="Maroon" title="c4">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>







<b>assume </b><a NAME="E1:40"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:40"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:40"/><i><font color="Green" title="E26">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<a NAME="E5:40"/>
<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="finseq_5.html#E3:40"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="finseq_5.html#T29" target="_self" title="FINSEQ_5:th.29">Th29</a></i>;<br/>
<a NAME="E6:40"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></i>;<br/>
<b>hence </b><a NAME="E7:40"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="finseq_5.html#E3:40"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="finseq_5.html#E4:40"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="finseq_5.html#T30" target="_self" title="FINSEQ_5:th.30">Th30</a></i>;<br/>


</div>
