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<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>assume </b><a NAME="E1:64"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x0</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">N</font> being    <a href="nfcont_1.html#M1" title="NFCONT_1:mode.1">Neighbourhood</a> of <font color="Maroon" title="c4">x0</font><b> such that </b><br/><a NAME="E3:64"/><i><font color="Green" title="E45">A2</font></i>: 
( <font color="Maroon" title="c5">N</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;  ex <font color="Olive" title="b1">L</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font></span>)</span> ex <font color="Olive" title="b2">R</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">T</font> st <br/>for <font color="Olive" title="b3">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">L</font> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x0</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="ndiff_1.html#D6" target="_self" title="NDIFF_1:def.6">Def6</a></i>;<br/>
<a NAME="E4:64"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 
<b>by </b><i><a class="ref" href="nfcont_1.html#T4" title="NFCONT_1:th.4">NFCONT_1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">s1</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="E1:64_1"/><i><font color="Green" title="E47">A4</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">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="Maroon" title="c6">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="Maroon" title="c6">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="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" title="c6">s1</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> ) )
 ;<br/><b>consider </b><font color="Maroon" title="c7">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:64_1"/><i><font color="Green" title="E48">A5</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">g</font> &amp; <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x0</font></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">g</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">N</font> )
 <b>by </b><i><a class="ref" href="nfcont_1.html#D3" title="NFCONT_1:def.3">NFCONT_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">s2</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="partfun2.html#K4" title="PARTFUN2:func.4">--&gt;</a> <font color="Maroon" title="c4">x0</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</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>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1</span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c9">s3</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E6:64_1"/><i><font color="Green" title="E49">A6</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  <font color="Maroon" title="c9">s3</font> <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">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E7:64_1"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c8">s2</font> is <a href="normsp_1.html#V4" title="NORMSP_1:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="ndiff_1.html#T21" target="_self" title="NDIFF_1:th.21">Th21</a></i>;<br/><i><font color="Green" title="E51">A8</font></i>:  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c8">s2</font> = 
<font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="ndiff_1.html#T21" target="_self" title="NDIFF_1:th.21">Th21</a></i>
<br/>.= 
<font color="Maroon" title="c4">x0</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><a NAME="E9:64_1"/><i><font color="Green" title="E52">A9</font></i>: 
<font color="Maroon" title="c9">s3</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c8">s2</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E6:64_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>;<br/><a NAME="E10:64_1"/><b>then </b><i><font color="Green" title="E53">A10</font></i>: 
<font color="Maroon" title="c9">s3</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#E1:64_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="ndiff_1.html#E7:64_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="normsp_1.html#T35" title="NORMSP_1:th.35">NORMSP_1:35</a></i>;<br/><i><font color="Green" title="E54">A11</font></i>:  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c9">s3</font> = 
<font color="Maroon" title="c4">x0</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x0</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:64_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="ndiff_1.html#E7:64_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="ndiff_1.html#E8:64_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="ndiff_1.html#E9:64_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="normsp_1.html#T43" title="NORMSP_1:th.43">NORMSP_1:43</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
;<br/><div><i><font color="Green" title="E55">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c10">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><a NAME="E2:64_1_3"/><i><font color="Green" title="E56">A13</font></i>: 
<font color="Maroon" title="c9">s3</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 ;<br/><a NAME="E3:64_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E6:64_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="ndiff_1.html#E2:64_1_3"><i><font color="Green" title="E56">A13</font></i></a></i>;<br/><a NAME="E4:64_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E5:64_1_3"/><b>then </b>
<font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c10">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="rlvect_1.html#T35" title="RLVECT_1:th.35">RLVECT_1:35</a></i>;<br/><b>hence </b><a NAME="E6:64_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:64_1"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c10">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="E14:64_1"/><i><font color="Green" title="E56">A14</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="c10">l</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"><font color="Maroon" title="c6">s1</font> <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#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x0</font></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">g</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:64_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="ndiff_1.html#E3:64_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="normsp_1.html#D11" title="NORMSP_1:def.11">NORMSP_1:def 11</a></i>;<br/><a NAME="E15:64_1"/><i><font color="Green" title="E57">A15</font></i>: 
( <font color="Maroon" title="c9">s3</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s3</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font> )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E10:64_1"><i><font color="Green" title="E53">A10</font></i></a>, <a class="txt" href="ndiff_1.html#E11:64_1"><i><font color="Green" title="E54">A11</font></i></a>, <a class="ref" href="lopban_3.html#T14" title="LOPBAN_3:th.14">LOPBAN_3:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c11">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><a NAME="E2:64_1_4"/><i><font color="Green" title="E58">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">s3</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 ;<br/><a NAME="E3:64_1_4"/>
<font color="Maroon" title="c9">s3</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:64_1_4"><i><font color="Green" title="E58">A16</font></i></a>, <a class="ref" href="lopban_3.html#D6" title="LOPBAN_3:def.6">LOPBAN_3:def 6</a></i>;<br/><b>hence </b><a NAME="E4:64_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="ndiff_1.html#E12:64_1"><i><font color="Green" title="E55">A12</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E17:64_1"/><b>then </b>
<font color="Maroon" title="c9">s3</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font> <a href="ndiff_1.html#NR1" title="NDIFF_1:NR.1">is_not_0</a> 
 <b>by </b><i><a class="ref" href="ndiff_1.html#T7" target="_self" title="NDIFF_1:th.7">Th7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c9">s3</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font> as  <a href="ndiff_1.html#V2" title="NDIFF_1:attr.2">convergent_to_0</a> <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="ndiff_1.html#E15:64_1"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="ndiff_1.html#D4" target="_self" title="NDIFF_1:def.4">Def4</a></i>;<br/><a NAME="E19:64_1"/>
<font color="Maroon" title="c8">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font> is    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c8">s2</font>
 <b>by </b><i><a class="ref" href="ndiff_1.html#T18" target="_self" title="NDIFF_1:th.18">Th18</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">c</font> = <font color="Maroon" title="c8">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font> as  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">V6</a> <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="ndiff_1.html#T19" target="_self" title="NDIFF_1:th.19">Th19</a></i>;<br/><a NAME="E21:64_1"/><i><font color="Green" title="E58">A17</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:64_1_5"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:64_1_5_1"/>
<font color="Maroon" title="c13">y</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="c12">c</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">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="E3:64_1_5_1"/><i><font color="Green" title="E59">A18</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6" title="NFCONT_1:th.6">NFCONT_1:6</a></i>;<br/>
<a NAME="E4:64_1_5_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E3:64_1_5_1"><i><font color="Green" title="E59">A18</font></i></a>, <a class="ref" href="lopban_3.html#D6" title="LOPBAN_3:def.6">LOPBAN_3:def 6</a></i>;<br/>
<a NAME="E5:64_1_5_1"/><b>then </b>
<font color="Maroon" title="c13">y</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="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E6:64_1_5_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:64_1_5"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E3:64_1_5"/><b>then </b><i><font color="Green" title="E59">A19</font></i>: 
<font color="Maroon" title="c13">y</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<font color="Maroon" title="c12">c</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0 = 
<font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" title="LOPBAN_3:def.6">LOPBAN_3:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E3:64_1_5"><i><font color="Green" title="E59">A19</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E5:64_1_5"/>
<font color="Maroon" title="c13">y</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="c12">c</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6" title="NFCONT_1:th.6">NFCONT_1:6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:64_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:64_1_6"/><i><font color="Green" title="E59">A20</font></i>: 
<font color="Maroon" title="c13">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">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="E3:64_1_6"/><i><font color="Green" title="E60">A21</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T6" title="NFCONT_1:th.6">NFCONT_1:6</a></i>;<br/>
<i><font color="Green" title="E61">A22</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c14">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c14">n</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E9:64_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="ndiff_1.html#T15" target="_self" title="NDIFF_1:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c8">s2</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="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" title="LOPBAN_3:def.6">LOPBAN_3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>
;<br/>
<a NAME="E5:64_1_6"/>
<font color="Maroon" title="c10">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E6:64_1_6"/><b>then </b><i><font color="Green" title="E62">A23</font></i>: 
<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="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c14">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x0</font></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">g</font>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E14:64_1"><i><font color="Green" title="E56">A14</font></i></a>, <a class="txt" href="ndiff_1.html#E4:64_1_6"><i><font color="Green" title="E61">A22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">y1</font> = <font color="Maroon" title="c13">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="ndiff_1.html#E1:64_1_6"><i><font color="Green" title="E59">A20</font></i></a></i>;<br/>
<a NAME="E8:64_1_6"/>
<font color="Maroon" title="c15">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> : <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">x0</font></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">g</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E3:64_1_6"><i><font color="Green" title="E60">A21</font></i></a>, <a class="txt" href="ndiff_1.html#E6:64_1_6"><i><font color="Green" title="E62">A23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:64_1_6"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:64_1"><i><font color="Green" title="E48">A5</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E23:64_1"/><b>then </b><i><font color="Green" title="E59">A24</font></i>: 
( <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="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</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="c12">c</font></span>)</span> 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> <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> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <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="c12">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:64"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="ndiff_1.html#E3:64"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="ndiff_1.html#E21:64_1"><i><font color="Green" title="E58">A17</font></i></a>, <a class="ref" href="ndiff_1.html#T39" target="_self" title="NDIFF_1:th.39">Th39</a></i>;<br/><div><i><font color="Green" title="E60">A25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:64_1_7"/><i><font color="Green" title="E61">A26</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">p</font>
 ;<br/><b>take </b><font color="Maroon" title="c14">n</font> = 0;<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="E2:64_1_7"/>
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 ;<br/><a NAME="E3:64_1_7"/>
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <b>by </b><i><a class="ref" href="nfcont_1.html#T4" title="NFCONT_1:th.4">NFCONT_1:4</a></i>;<br/><a NAME="E4:64_1_7"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">c</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>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:64"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="ndiff_1.html#E21:64_1"><i><font color="Green" title="E58">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><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> <font color="Maroon" title="c12">c</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#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">c</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#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c15">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="ref" href="lopban_3.html#D6" title="LOPBAN_3:def.6">LOPBAN_3:def 6</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="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</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/>.= 
0
<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="E6:64_1_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="c12">c</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#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c13">p</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:64_1_7"><i><font color="Green" title="E61">A26</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E25:64_1"/><b>then </b><i><font color="Green" title="E61">A27</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c12">c</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/><a NAME="E26:64_1"/><b>then </b><i><font color="Green" title="E62">A28</font></i>: 
 <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="c12">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">x0</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E24:64_1"><i><font color="Green" title="E60">A25</font></i></a>, <a class="ref" href="normsp_1.html#D11" title="NORMSP_1:def.11">NORMSP_1:def 11</a></i>;<br/><a NAME="E27:64_1"/><i><font color="Green" title="E63">A29</font></i>: 
<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> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <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="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</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="c12">c</font></span>)</span> 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#E23:64_1"><i><font color="Green" title="E59">A24</font></i></a>, <a class="txt" href="ndiff_1.html#E25:64_1"><i><font color="Green" title="E61">A27</font></i></a>, <a class="ref" href="normsp_1.html#T34" title="NORMSP_1:th.34">NORMSP_1:34</a></i>;<br/><i><font color="Green" title="E64">A30</font></i>:  <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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<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="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <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="c12">c</font></span>)</span></span>)</span> = 
<span class="p1">(<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> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E23:64_1"><i><font color="Green" title="E59">A24</font></i></a>, <a class="txt" href="ndiff_1.html#E25:64_1"><i><font color="Green" title="E61">A27</font></i></a>, <a class="txt" href="ndiff_1.html#E26:64_1"><i><font color="Green" title="E62">A28</font></i></a>, <a class="ref" href="normsp_1.html#T42" title="NORMSP_1:th.42">NORMSP_1:42</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">x0</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">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>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<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="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <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="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <span class="p3">(<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="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="c12">c</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<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="c12">c</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="c12">c</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>
<br/>.= 
<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> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c12">c</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<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="c12">c</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></i>
<br/>.= 
<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> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<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>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<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="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>
;<br/></div><b>end;</b></div><a NAME="E30:64_1"/><b>then </b><i><font color="Green" title="E65">A31</font></i>: 
<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> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <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="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</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="c12">c</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <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="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">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>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E9:64_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="ndiff_1.html#T15" target="_self" title="NDIFF_1:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c8">s2</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="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span>
<b>by </b><i><a class="ref" href="lopban_3.html#D6" title="LOPBAN_3:def.6">LOPBAN_3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <font color="Maroon" title="c8">s2</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D5" title="NORMSP_1:def.5">NORMSP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D6" title="NORMSP_1:def.6">NORMSP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">s2</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c10">l</font></span>)</span></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">S</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c6">s1</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c13">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s1</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c13">n</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><b>then </b><i><font color="Green" title="E66">A32</font></i>: <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> <span class="p3">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</a> <font color="Maroon" title="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K4" title="NORMSP_1:func.4">-</a> <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="c12">c</font></span>)</span></span>)</span> <a href="normsp_1.html#K3" title="NORMSP_1:func.3">+</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="c12">c</font></span>)</span> = 
<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="c6">s1</font> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E30:64_1"><i><font color="Green" title="E65">A31</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>
<br/>.= 
<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="c6">s1</font></span>)</span> <a href="lopban_3.html#K2" title="LOPBAN_3:func.2">^\</a> <font color="Maroon" title="c10">l</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E1:64_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="ndiff_1.html#T29" target="_self" title="NDIFF_1:th.29">Th29</a></i>
;<br/><b>hence </b><a NAME="E33:64_1"/>
<font color="Maroon" title="c3">f</font> <a href="nfcont_1.html#K2" title="NFCONT_1:func.2">*</a> <font color="Maroon" title="c6">s1</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#E27:64_1"><i><font color="Green" title="E63">A29</font></i></a>, <a class="ref" href="lopban_3.html#T15" title="LOPBAN_3:th.15">LOPBAN_3:15</a></i>;<br/><b>thus </b><a NAME="E34:64_1"/>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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="c6">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E27:64_1"><i><font color="Green" title="E63">A29</font></i></a>, <a class="txt" href="ndiff_1.html#E28:64_1"><i><font color="Green" title="E64">A30</font></i></a>, <a class="txt" href="ndiff_1.html#E32:64_1"><i><font color="Green" title="E66">A32</font></i></a>, <a class="ref" href="lopban_3.html#T16" title="LOPBAN_3:th.16">LOPBAN_3:16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:64"/>
<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>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E3:64"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="ndiff_1.html#E4:64"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="ndiff_1.html#T32" target="_self" title="NDIFF_1:th.32">Th32</a></i>;<br/>


</div>
