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<b>let </b><font color="Maroon">RNS</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">RNS</font>;<br/>





<b>assume </b><a NAME="E1:37"/>
<font color="Maroon">S</font> is <a href="normsp_1.html#V4" target="_self">convergent</a>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RNS</font><b> such that </b><br/><a NAME="E3:37"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font>
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">h</font> = <font color="Maroon">g</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="normsp_1.html#D9" target="_self">NORMSP_1:def 9</a><br/>

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

<b>assume </b><a NAME="E4:37"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:37"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="normsp_1.html#E8"><i><font color="Green">Lemma32</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">k</font> = <font color="Maroon">m1</font>;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E7:37"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E8:37"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="normsp_1.html#E9"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> = 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</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">S</font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">RNS</font>)</span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</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">S</font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</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">S</font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">g</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</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">S</font> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">h</font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47">RLVECT_1:47</a></i>
;<br/>
<b>hence </b><a NAME="E10:37"/>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</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">S</font> <a href="normsp_1.html#K5" target="_self">-</a> <font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">h</font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>


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