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<b>let </b><font color="Maroon">X</font> be  non <a href="realset2.html#V3">trivial</a> <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/>

<b>consider </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:39"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a></i>;<br/>
<a NAME="E3:39"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</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="E4:39"/><b>then </b><i><font color="Green">E47</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">v</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/>
<b>set </b><font color="Maroon">a</font> = <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</font></span><a href="normsp_1.html#K1">.||</a></span>;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> ;<br/>
<i><font color="Green">E48</font></i>:  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <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">v</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T15">ABSVALUE:15</a></i>
<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</font></span><a href="normsp_1.html#K1">.||</a></span></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>thus </b><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">w</font></span><a href="normsp_1.html#K1">.||</a></span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">v</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"><font color="Maroon">v</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/>.= 
1
<b>by </b><i>, , <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
;<br/>


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