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<b>let </b><font color="Maroon">X</font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>





<a NAME="E1:73"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <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> = 
<font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>)
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>
;<br/>
<a NAME="E3:73"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span>
 
<b>by </b><i><a class="ref" href="csspace.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E4:73"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E5:73"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span>
 
;<br/>
<a NAME="E6:73"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T47">RLVECT_1:47</a></i>;<br/>
<a NAME="E7:73"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:73"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E9:73"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span>
 
;<br/>
<b>hence </b><a NAME="E10:73"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K5">-</a> <span class="p2"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span>
 <b>by </b><i><a class="ref" href="csspace.html#T3" target="_self">Th3</a>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>


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