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

<b>let </b><font color="Maroon">A</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">B</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">C</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:89"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font>
 <b>and </b><br/><a NAME="E2:89"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">C</font> is    <a href="kurato_2.html#M1" target="_self">subsequence</a> of <font color="Maroon">B</font>
 ;<br/>

<b>consider </b><font color="Maroon">NS</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E4:89"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">NS</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E4:89"><i><font color="Green">E54</font></i></a>, </i>;<br/>
<b>set </b><font color="Maroon">D</font> = <font color="Maroon">A</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">NS</font>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">A</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">NS</font> as   <a href="kurato_2.html#NM1">SetSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as    <a href="kurato_2.html#M1" target="_self">subsequence</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">D</font>
;<br/>

<a NAME="E7:89"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">D</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<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/>

<a NAME="E1:89_1"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">NS</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:89_1"/>
<font color="Maroon">D</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:89_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E1:89_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E3:89_1_1"/>
<font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:89_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">NS</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E2:89_1_1"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:89_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="kurato_2.html#E1:89_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:89_1"/>
<font color="Maroon">D</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:89"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">D</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font>
 ;<br/>


</div>
