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<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>;<br/>







<b>thus </b><a NAME="E1:41"/>
( <font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#R1" title="NFCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c4">x0</font> implies ( <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font> &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Olive" title="b1">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> ) holds <br/>( <font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font></span>)</span> ) ) ) )
 <b>by </b><i><a class="ref" href="nfcont_1.html#D9" title="NFCONT_1:def.9">NFCONT_1:def 9</a></i>;<br/>

<b>assume </b><a NAME="E2:41"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font> &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Olive" title="b1">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> ) holds <br/>( <font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Olive" title="b1">s1</font></span>)</span> ) ) )
 ;<br/>

<b>hence </b><a NAME="E3:41"/>
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="nfcont_1.html#D9" title="NFCONT_1:def.9">NFCONT_1:def 9</a><br/>

<b>let </b><font color="Maroon" title="c5">s2</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><a NAME="E4:41"/><i><font color="Green" title="E32">A2</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">s2</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c5">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1"/>
(  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font> or for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">N</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:41_1_1_1"/><i><font color="Green" title="E33">A3</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 ;<br/><a NAME="E4:41_1_1_1"/><i><font color="Green" title="E34">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:41_1_1_1_1"/>
<font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E3:41_1_1_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E2:41_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 <b>by </b><i><a class="ref" href="lopban_3.html#D6" title="LOPBAN_3:def.6">LOPBAN_3:def 6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:41_1_1_1"/>
<font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c6">N</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c5">s2</font>
 <b>by </b><i><a class="ref" href="lopban_3.html#T13" title="LOPBAN_3:th.13">LOPBAN_3:13</a></i>;<br/><a NAME="E6:41_1_1_1"/><b>then </b><i><font color="Green" title="E35">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">s2</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T7" title="NFCONT_1:th.7">NFCONT_1:7</a></i>;<br/><div><i><font color="Green" title="E36">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:41_1_1_1_2"/><i><font color="Green" title="E37">A7</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>take </b><font color="Maroon" title="c8">n</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>let </b><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E2:41_1_1_1_2"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 ;<br/><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c6">N</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E4:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="ndiff_1.html#E6:41_1_1_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="nfcont_1.html#T8" title="NFCONT_1:th.8">NFCONT_1:8</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E4:41_1_1_1"><i><font color="Green" title="E34">A4</font></i></a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="normsp_1.html#T5" title="NORMSP_1:th.5">NORMSP_1:5</a></i>
;<br/><b>hence </b><a NAME="E4:41_1_1_1_2"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c6">N</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:41_1_1_1_2"><i><font color="Green" title="E37">A7</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:41_1_1_1"/><b>then </b><i><font color="Green" title="E37">A8</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c6">N</font></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9" title="NORMSP_1:def.9">NORMSP_1:def 9</a></i>;<br/><a NAME="E9:41_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c6">N</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font></span>)</span> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c6">N</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="ndiff_1.html#T29" target="_self" title="NDIFF_1:th.29">Th29</a></i>;<br/><a NAME="E10:41_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font></span>)</span> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c6">N</font></span>)</span> )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E7:41_1_1_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="ndiff_1.html#E8:41_1_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="lopban_3.html#T15" title="LOPBAN_3:th.15">LOPBAN_3:15</a>, <a class="ref" href="normsp_1.html#D11" title="NORMSP_1:def.11">NORMSP_1:def 11</a></i>;<br/><b>hence </b><a NAME="E11:41_1_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="lopban_3.html#T14" title="LOPBAN_3:th.14">LOPBAN_3:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2"/><i><font color="Green" title="E33">A9</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">m1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:41_1_1_2"/><i><font color="Green" title="E34">A10</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m1</font> &amp; <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c5">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>;<br/><a NAME="E4:41_1_1_2"/><i><font color="Green" title="E35">A11</font></i>: 
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:41_1_1_2"><i><font color="Green" title="E34">A10</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">M</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:41_1_1_2"/><i><font color="Green" title="E36">A12</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">M</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c7">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="ndiff_1.html#E4:41_1_1_2"><i><font color="Green" title="E35">A11</font></i></a>);<br/></i><div><i><font color="Green" title="E37">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>consider </b><font color="Maroon" title="c9">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:41_1_1_2_1"/><i><font color="Green" title="E38">A14</font></i>: 
( <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:41_1_1_2"><i><font color="Green" title="E33">A9</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c10">m</font> = <font color="Maroon" title="c9">m</font>;<br/><b>thus </b><a NAME="E3:41_1_1_2_1"/>
( <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">m</font> &amp; <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:41_1_1_2_1"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> holds <br/>( <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> &amp; ( for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">k</font> &amp; <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> holds <br/><font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> ) );<br/><a NAME="E8:41_1_1_2"/><i><font color="Green" title="E38">A15</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">n</font>, <font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> $1 &amp; <font color="Maroon" title="c5">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> );<br/>
<a NAME="E1:41_1_1_2_2"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E7:41_1_1_2"><i><font color="Green" title="E37">A13</font></i></a></i>;<br/>
<a NAME="E2:41_1_1_2_2"/><b>then </b><i><font color="Green" title="E39">A16</font></i>: 
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">m</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c10">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:41_1_1_2_2"/><i><font color="Green" title="E40">A17</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">l</font>] &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="ndiff_1.html#E2:41_1_1_2_2"><i><font color="Green" title="E39">A16</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c11">l</font> = <font color="Maroon" title="c10">l</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">l</font>
;<br/>

<b>thus </b><a NAME="E6:41_1_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">l</font>]
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:41_1_1_2_2"><i><font color="Green" title="E40">A17</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c8">M'</font> = <font color="Maroon" title="c7">M</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:41_1_1_2"/><i><font color="Green" title="E39">A18</font></i>: 
( <font color="Maroon" title="c9">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">M'</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c9">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c9">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="ndiff_1.html#E8:41_1_1_2"><i><font color="Green" title="E38">A15</font></i></a>);<br/></i><a NAME="E12:41_1_1_2"/><i><font color="Green" title="E40">A19</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E13:41_1_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">F</font> = <font color="Maroon" title="c9">F</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="txt" href="ndiff_1.html#E12:41_1_1_2"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/><div><i><font color="Green" title="E41">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:41_1_1_2_3"/>
<font color="Maroon" title="c10">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E12:41_1_1_2"><i><font color="Green" title="E40">A19</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:41_1_1_2_3"/>
<font color="Maroon" title="c10">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="ndiff_1.html#E12:41_1_1_2"><i><font color="Green" title="E40">A19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:41_1_1_2_4"/>
( <font color="Maroon" title="c10">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c10">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E15:41_1_1_2"><i><font color="Green" title="E41">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E2:41_1_1_2_4"/>
<font color="Maroon" title="c10">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E11:41_1_1_2"><i><font color="Green" title="E39">A18</font></i></a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">F</font> = <font color="Maroon" title="c10">F</font> as  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a> <b>by </b><i><a class="ref" href="seqm_3.html#D11" title="SEQM_3:def.11">SEQM_3:def 11</a></i>;<br/><a NAME="E18:41_1_1_2"/><i><font color="Green" title="E42">A21</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> holds <br/> ex <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_5"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( <font color="Maroon" title="c5">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $1 ) );<br/>
<b>assume </b><a NAME="E1:41_1_1_2_5"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 ;<br/>

<a NAME="E2:41_1_1_2_5"/><b>then </b><i><font color="Green" title="E43">A22</font></i>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c12">M1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:41_1_1_2_5"/><i><font color="Green" title="E44">A23</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c12">M1</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c12">M1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="ndiff_1.html#E2:41_1_1_2_5"><i><font color="Green" title="E43">A22</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">M1</font> &amp; <font color="Maroon" title="c5">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> &amp;  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E5:41_1_1_2_5"/><i><font color="Green" title="E45">A24</font></i>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_5_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c7">M</font>
;<br/>

<a NAME="E1:41_1_1_2_5_1"/><i><font color="Green" title="E46">A25</font></i>: 
<font color="Maroon" title="c7">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">M1</font>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E6:41_1_1_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="txt" href="ndiff_1.html#E4:41_1_1_2_5"><i><font color="Green" title="E44">A23</font></i></a></i>;<br/>
<a NAME="E2:41_1_1_2_5_1"/>
<font color="Maroon" title="c7">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">M1</font>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:41_1_1_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="ndiff_1.html#E4:41_1_1_2_5"><i><font color="Green" title="E44">A23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_1_2_5_1"/>
<font color="Maroon" title="c7">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">M1</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:41_1_1_2_5_1"><i><font color="Green" title="E46">A25</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E4:41_1_1_2_5_1"/>
<font color="Maroon" title="c5">s2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E6:41_1_1_2"><i><font color="Green" title="E36">A12</font></i></a></i>;<br/>

<b>take </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
;<br/>

<b>thus </b><a NAME="E5:41_1_1_2_5_1"/>
<font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">M</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E11:41_1_1_2"><i><font color="Green" title="E39">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:41_1_1_2_5"/><i><font color="Green" title="E46">A26</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">M1</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c13">MX</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:41_1_1_2_5"/><i><font color="Green" title="E47">A27</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c13">MX</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">MX</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="ndiff_1.html#E6:41_1_1_2_5"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="ndiff_1.html#E5:41_1_1_2_5"><i><font color="Green" title="E45">A24</font></i></a>);<br/></i>
<a NAME="E9:41_1_1_2_5"/><i><font color="Green" title="E48">A28</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c13">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">M1</font> holds <br/><font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_5_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c14">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:41_1_1_2_5_2"/><i><font color="Green" title="E49">A29</font></i>: 
( <font color="Maroon" title="c13">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">k</font> &amp; <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">M1</font> &amp; <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_5_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_5_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_2_5_2_1_1"/>
(  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> or for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_5_2_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_2_5_2_1_1_1"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_2_5_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ndiff_1.html#E8:41_1_1_2_5"><i><font color="Green" title="E47">A27</font></i></a>, <a class="txt" href="ndiff_1.html#E2:41_1_1_2_5_2"><i><font color="Green" title="E49">A29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_5_2_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2_5_2_1_1_2"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_2_5_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:41_1_1_2_5"><i><font color="Green" title="E44">A23</font></i></a>, <a class="txt" href="ndiff_1.html#E2:41_1_1_2_5_2"><i><font color="Green" title="E49">A29</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:41_1_1_2_5_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c14">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:41_1_1_2_5"/><i><font color="Green" title="E49">A30</font></i>: 
<font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MX</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E8:41_1_1_2_5"><i><font color="Green" title="E47">A27</font></i></a></i>;<br/>
<a NAME="E12:41_1_1_2_5"/><i><font color="Green" title="E50">A31</font></i>: 
( <font color="Maroon" title="c13">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c13">MX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> holds <br/><font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:41_1_1_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="ndiff_1.html#E11:41_1_1_2_5"><i><font color="Green" title="E49">A30</font></i></a></i>;<br/>
<a NAME="E13:41_1_1_2_5"/>
<font color="Maroon" title="c12">M1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E14:41_1_1_2_5"/><b>then </b><i><font color="Green" title="E51">A32</font></i>: 
<font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">M1</font>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:41_1_1_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="ndiff_1.html#E4:41_1_1_2_5"><i><font color="Green" title="E44">A23</font></i></a>, <a class="txt" href="ndiff_1.html#E8:41_1_1_2_5"><i><font color="Green" title="E47">A27</font></i></a>, <a class="txt" href="ndiff_1.html#E11:41_1_1_2_5"><i><font color="Green" title="E49">A30</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1_2_5_3"/>
<font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">M1</font>
 ;<br/><a NAME="E2:41_1_1_2_5_3"/><b>then </b>
<font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">M1</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E14:41_1_1_2_5"><i><font color="Green" title="E51">A32</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:41_1_1_2_5_3"/>
contradiction
 <b>by </b><i><a class="txt" href="ndiff_1.html#E9:41_1_1_2_5"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="ndiff_1.html#E12:41_1_1_2_5"><i><font color="Green" title="E50">A31</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:41_1_1_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:41_1_1_2_5"><i><font color="Green" title="E44">A23</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:41_1_1_2"/><i><font color="Green" title="E43">A33</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_6"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>;<br/>
<a NAME="E1:41_1_1_2_6"/><i><font color="Green" title="E44">A34</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E6:41_1_1_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="txt" href="ndiff_1.html#E11:41_1_1_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="ref" href="lopban_3.html#T7" title="LOPBAN_3:th.7">LOPBAN_3:7</a></i>;<br/>
<a NAME="E2:41_1_1_2_6"/><i><font color="Green" title="E45">A35</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:41_1_1_2_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>
 ;<br/>

<a NAME="E2:41_1_1_2_6_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E11:41_1_1_2"><i><font color="Green" title="E39">A18</font></i></a></i>;<br/>
<a NAME="E3:41_1_1_2_6_1"/><b>then </b>
<font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>
 
;<br/>
<b>hence </b><a NAME="E4:41_1_1_2_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>
 <b>by </b><i><a class="ref" href="lopban_3.html#T7" title="LOPBAN_3:th.7">LOPBAN_3:7</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:41_1_1_2_6"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="ndiff_1.html#E1:41_1_1_2_6"><i><font color="Green" title="E44">A34</font></i></a>, <a class="txt" href="ndiff_1.html#E2:41_1_1_2_6"><i><font color="Green" title="E45">A35</font></i></a>);</i><br/>


</div><b>end;</b></div><a NAME="E20:41_1_1_2"/><i><font color="Green" title="E44">A36</font></i>: 
<font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c5">s2</font>
 <b>by </b><i><a class="ref" href="bhsp_3.html#D4" title="BHSP_3:def.4">BHSP_3:def 4</a></i>;<br/><a NAME="E21:41_1_1_2"/><b>then </b><i><font color="Green" title="E45">A37</font></i>: 
<font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="lopban_3.html#T11" title="LOPBAN_3:th.11">LOPBAN_3:11</a></i>;<br/><a NAME="E22:41_1_1_2"/><i><font color="Green" title="E46">A38</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">s2</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E20:41_1_1_2"><i><font color="Green" title="E44">A36</font></i></a>, <a class="ref" href="nfcont_1.html#T7" title="NFCONT_1:th.7">NFCONT_1:7</a></i>;<br/><a NAME="E23:41_1_1_2"/><b>then </b><i><font color="Green" title="E47">A39</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E24:41_1_1_2"/>
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="ndiff_1.html#E20:41_1_1_2"><i><font color="Green" title="E44">A36</font></i></a>, <a class="ref" href="lopban_3.html#T12" title="LOPBAN_3:th.12">LOPBAN_3:12</a></i>;<br/><a NAME="E25:41_1_1_2"/><b>then </b><i><font color="Green" title="E48">A40</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:41"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="ndiff_1.html#E19:41_1_1_2"><i><font color="Green" title="E43">A33</font></i></a>, <a class="txt" href="ndiff_1.html#E21:41_1_1_2"><i><font color="Green" title="E45">A37</font></i></a>, <a class="txt" href="ndiff_1.html#E23:41_1_1_2"><i><font color="Green" title="E47">A39</font></i></a></i>;<br/><div><i><font color="Green" title="E49">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:41_1_1_2_7"/><i><font color="Green" title="E50">A42</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">p</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:41_1_1_2_7"/><i><font color="Green" title="E51">A43</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E25:41_1_1_2"><i><font color="Green" title="E48">A40</font></i></a>, <a class="ref" href="normsp_1.html#D11" title="NORMSP_1:def.11">NORMSP_1:def 11</a></i>;<br/><b>take </b><font color="Maroon" title="c14">k</font> = <font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">n</font>;<br/><b>let </b><font color="Maroon" title="c15">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E4:41_1_1_2_7"/><i><font color="Green" title="E52">A44</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_2_7_1_1"/>
( <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font> or <font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_7_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_2_7_1_1_1"/><i><font color="Green" title="E53">A45</font></i>: 
<font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font>
 ;<br/><div class="add"><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E4:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="nfcont_1.html#T8" title="NFCONT_1:th.8">NFCONT_1:8</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:41_1_1_2_7_1_1_1"><i><font color="Green" title="E53">A45</font></i></a>, <a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="normsp_1.html#T5" title="NORMSP_1:th.5">NORMSP_1:5</a></i>
;<br/><b>hence </b><a NAME="E3:41_1_1_2_7_1_1_1"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:41_1_1_2_7"><i><font color="Green" title="E50">A42</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_7_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">l</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:41_1_1_2_7_1_1_2"/><i><font color="Green" title="E53">A46</font></i>: 
<font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">l</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E18:41_1_1_2"><i><font color="Green" title="E42">A21</font></i></a></i>;<br/><a NAME="E4:41_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">l</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:41_1_1_2_7"><i><font color="Green" title="E52">A44</font></i></a>, <a class="txt" href="ndiff_1.html#E3:41_1_1_2_7_1_1_2"><i><font color="Green" title="E53">A46</font></i></a>, <a class="ref" href="seqm_3.html#T7" title="SEQM_3:th.7">SEQM_3:7</a></i>;<br/><a NAME="E5:41_1_1_2_7_1_1_2"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c16">l</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:41_1_1_2_7"><i><font color="Green" title="E51">A43</font></i></a></i>;<br/><a NAME="E6:41_1_1_2_7_1_1_2"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="lopban_3.html#K4" title="LOPBAN_3:func.4">*</a> <font color="Maroon" title="c11">F</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c16">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="ndiff_1.html#E22:41_1_1_2"><i><font color="Green" title="E46">A38</font></i></a>, <a class="ref" href="nfcont_1.html#T8" title="NFCONT_1:th.8">NFCONT_1:8</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:41_1_1_2_7_1_1_2"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:41_1_1_2_7_1_1_2"><i><font color="Green" title="E53">A46</font></i></a>, <a class="ref" href="lopban_3.html#T7" title="LOPBAN_3:th.7">LOPBAN_3:7</a></i>;<br/><b>hence </b><a NAME="E8:41_1_1_2_7_1_1_2"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E4:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="nfcont_1.html#T8" title="NFCONT_1:th.8">NFCONT_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:41_1_1_2_7"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">p</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E27:41_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="normsp_1.html#D9" title="NORMSP_1:def.9">NORMSP_1:def 9</a></i>;<br/><b>hence </b><a NAME="E28:41_1_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font></span>)</span>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E26:41_1_1_2"><i><font color="Green" title="E49">A41</font></i></a>, <a class="ref" href="normsp_1.html#D11" title="NORMSP_1:def.11">NORMSP_1:def 11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:41"/>
( <font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c5">s2</font></span>)</span> )
 ;<br/>


</div>
