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<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>





<a NAME="E1:17"/><i><font color="Green">E15</font></i>: 
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T12" target="_self">Th12</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/>
<a NAME="E3:17"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:17"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E5:17"/><b>then </b>
<span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E6:17"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E7:17"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
 
;<br/>
<b>hence </b><a NAME="E8:17"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="normsp_1.html#K1" target="_self">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="normsp_1.html#K1" target="_self">.||</a></span>
 <b>by </b><i><a class="txt" href="normsp_1.html#E1:17"><i><font color="Green">E15</font></i></a>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>


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