<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="realset2.html#V3">trivial</a> <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">T</font> be  non <a href="realset2.html#V3">trivial</a> <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font>
 ;<br/>

<b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E3:11"/>
<font color="Maroon">R</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="ndiff_1.html#D5">NDIFF_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:11"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E6:11"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> &amp; <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">z</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">z</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="ndiff_1.html#T26">NDIFF_1:26</a></i>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1_1"/>
( <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> or <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_1"><b>case </b></a><a NAME="E1:11_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><a NAME="E2:11_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/><a NAME="E3:11_1_1_1_1"/><i><font color="Green">E46</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/><a NAME="E4:11_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">h</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:11_1_1_1_1"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">h</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:11_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">h</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span>
 ;<br/><a NAME="E7:11_1_1_1_1"/><b>then </b>
1 <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">h</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>hence </b><a NAME="E8:11_1_1_1_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">h</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_2"><b>case </b></a><a NAME="E1:11_1_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 ;<br/><div class="add"><a NAME="E2:11_1_1_1_2"/><i><font color="Green">E48</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i><a class="ref" href="normsp_1.html#T8">NORMSP_1:8</a></i>;<br/><a NAME="E3:11_1_1_1_2"/>
0 <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E4:11_1_1_1_2"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">h</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span>
 <b>by </b><i>, , <a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:11_1"/>
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">h</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K1">.||</a></span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:11"/>
( <font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">h</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">h</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> holds <br/><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">h</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Olive">h</font></span><a href="normsp_1.html#K1">.||</a></span> ) )
 <b>by </b><i/>;<br/>


</div>
