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<b>let </b><font color="Maroon" title="c1">X</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">s1</font> being  <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">s</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Olive" title="b2">s1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b1">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">s1</font></span><br/><b>let </b><font color="Maroon" title="c2">s</font>, <font color="Maroon" title="c3">s1</font> be   <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">s</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c3">s1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c2">s</font> implies <font color="Maroon" title="c2">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">s1</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:148"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c2">s</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>and </b><br/><a NAME="E2:148"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c3">s1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c2">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">s1</font></span><br/>

<b>consider </b><font color="Maroon" title="c4">N</font> being  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> <a href="nat_1.html#NM2" title="NAT_1:NM.2">sequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:148"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c3">s1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">s</font> <a href="valued_0.html#K2" title="VALUED_0:func.2">*</a> <font color="Maroon" title="c4">N</font>
 <b>by </b><i><a class="txt" href="#E2:148"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="valued_0.html#D17" target="_self" title="VALUED_0:def.17">Def17</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D9" title="FUNCT_2:def.9">FUNCT_2:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span><br/>

<b>thus </b><font color="Maroon" title="c3">s1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font> = 
<font color="Maroon" title="c2">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:148"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="txt" href="#E1:148"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="valued_0.html#T23" target="_self" title="VALUED_0:th.23">Th23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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