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<b>let </b><font color="Maroon" title="c1">X</font> be    <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">s1</font>, <font color="Olive" title="b2">s2</font>, <font color="Olive" title="b3">s3</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">s1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Olive" title="b2">s2</font> &amp; <font color="Olive" title="b2">s2</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Olive" title="b3">s3</font> holds <br/><font color="Olive" title="b1">s1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Olive" title="b3">s3</font></span><br/><b>let </b><font color="Maroon" title="c2">s1</font>, <font color="Maroon" title="c3">s2</font>, <font color="Maroon" title="c4">s3</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">s1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c3">s2</font> &amp; <font color="Maroon" title="c3">s2</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c4">s3</font> implies <font color="Maroon" title="c2">s1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c4">s3</font> )</span><br/>







<b>given </b><font color="Maroon" title="c5">N1</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><a NAME="E2:136"/><i><font color="Green" title="E30">G1</font></i>: 
<font color="Maroon" title="c2">s1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">s2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">N1</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="valued_0.html#D17" target="_self" title="VALUED_0:def.17">VALUED_0:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">s2</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c4">s3</font> or <font color="Maroon" title="c2">s1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c4">s3</font> )</span><br/>

<b>given </b><font color="Maroon" title="c6">N2</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><a NAME="E4:136"/><i><font color="Green" title="E31">G2</font></i>: 
<font color="Maroon" title="c3">s2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">s3</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">N2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="valued_0.html#D17" target="_self" title="VALUED_0:def.17">VALUED_0:def 17</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">s1</font> is    <a href="valued_0.html#M1" title="VALUED_0:mode.1">subsequence</a> of <font color="Maroon" title="c4">s3</font></span><br/>

<b>take </b>
<font color="Maroon" title="c6">N2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">N1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="valued_0.html#D17" target="_self" title="VALUED_0:def.17">VALUED_0:def 17</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">s1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">s3</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">N1</font></span>)</span></span><br/>

<b>thus </b><a NAME="E5:136"/>
<font color="Maroon" title="c2">s1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">s3</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">N2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">N1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:136"><i><font color="Green" title="E30">G1</font></i></a>, <a class="txt" href="#E4:136"><i><font color="Green" title="E31">G2</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</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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