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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>, <font color="Maroon">G</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G</font> is    <a href="kurato_2.html#M1" target="_self">subsequence</a> of <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green">E49</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">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</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="E4:62"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">NS</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:62"/><i><font color="Green">E56</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">G</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</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="62_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="E1:62_1"/>
 <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/>
<b>then </b><font color="Maroon">G</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">F</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="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:62_1"/>
<font color="Maroon">G</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:62"/><i><font color="Green">E57</font></i>: 
(  <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp;  <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:62"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:62"/>
<font color="Maroon">G</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:62"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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