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<span class="kw">set </span><font color="Maroon" title="c4">N</font> = <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font>;<br/>
<a NAME="E1:137_1_1"/>
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">e1</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="valued_0.html#D13" target="_self" title="VALUED_0:def.13">VALUED_0:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e2</font> being   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp; <font color="Olive" title="b1">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp; <font color="Maroon" title="c5">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e2</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">e2</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp; <font color="Maroon" title="c6">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp; <font color="Maroon" title="c5">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e2</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e2</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:137_1_1_1"/>
( <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> &amp; <font color="Maroon" title="c6">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e2</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c5">e1</font>, <font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c6">e2</font> as    <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/>
<span class="lab"><font color="Green" title="E31">A1</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">j</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E4:137_1_1_1"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c5">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">e2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e2</font></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" title="NAT_1:def.3">NAT_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:137_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:137_1_1_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E4:137_1_1_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">N</font> = <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font> as   <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> ;<br/>
<span class="kw">thus </span><a NAME="E3:137_1_1"/>
<font color="Maroon" title="c2">s</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c2">s</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c5">N</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">s</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">N</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c2">s</font> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">^\</a> <font color="Maroon" title="c3">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></span>
<br/>.= 
<font color="Maroon" title="c2">s</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">N</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T50" title="NAT_1:th.50">NAT_1:50</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
