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<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">seq1</font> is <a href="funct_1.html#V3">constant</a>
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">seq2</font> is <a href="funct_1.html#V3">constant</a>
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">seq</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq1</font> <a href="bhsp_1.html#K8" target="_self">+</a> <font color="Maroon">seq2</font>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:57"/><i><font color="Green">E68</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>  holds  <font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#D4">NORMSP_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:57"/><i><font color="Green">E69</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>  holds  <font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="normsp_1.html#D4">NORMSP_1:def 4</a></i>;<br/>
<b>take </b><font color="Maroon">z</font> = <font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="normsp_1.html#D4">NORMSP_1:def 4</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><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>thus </b><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="normsp_1.html#D5">NORMSP_1:def 5</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_1.html#E3"><i><font color="Green">Lemma36</font></i></a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:57"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">seq</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/>


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