<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:56"/><i><font color="Green">E32</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_5.html#E2:56"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:56_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:56_1_1_1_1"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E1:56"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E2:56_1_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:56_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:56_1_1_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="txt" href="rusub_5.html#E4:56_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="rusub_5.html#E4:56_1_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E7:56_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E5:56_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:56_1_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T32">BHSP_1:32</a>, <a class="ref" href="bhsp_1.html#T34">BHSP_1:34</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="bhsp_1.html#K3">.||</a></span>;<br/>
<b>take </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>;<br/>

<a NAME="E9:56_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="rusub_5.html#E4:56"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rusub_5.html#E8:56_1_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T33">BHSP_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="txt" href="rusub_5.html#E9:56_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T82">XCMPLX_1:82</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="rusub_5.html#E8:56_1_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xcmplx_1.html#T51">XCMPLX_1:51</a></i>
;<br/>
<b>hence </b><a NAME="E11:56_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_5.html#E4:56"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rusub_5.html#E7:56_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rusub_5.html#E9:56_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rlvect_1.html#T24">RLVECT_1:24</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:56_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> = 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="txt" href="rusub_5.html#E1:56_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="rlvect_1.html#T27">RLVECT_1:27</a></i>
<br/>.= 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T37">BHSP_1:37</a></i>
;<br/><a NAME="E6:56_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="rusub_5.html#E4:56_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E7:56_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="bhsp_2.html#D5">BHSP_2:def 5</a></i>;<br/><b>hence </b><a NAME="E8:56_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:56_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="rusub_5.html#E4:56_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span>;<br/><a NAME="E2:56_1_1_2"/><i><font color="Green">E34</font></i>: 
( <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:56_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="bhsp_1.html#T32">BHSP_1:32</a>, <a class="ref" href="bhsp_1.html#T34">BHSP_1:34</a></i>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>;<br/><i><font color="Green">E35</font></i>: <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p0"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> = 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p0"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p0"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T49">RLVECT_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p5"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="ref" href="bhsp_1.html#T33">BHSP_1:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span>

;<br/><a NAME="E4:56_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="rusub_5.html#E4:56"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rusub_5.html#E2:56_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>then </b><i><font color="Green">E36</font></i>: <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p0"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span>
<b>by </b><i><a class="txt" href="rusub_5.html#E3:56_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span> <a href="real_1.html#K6">/</a> <span class="p2"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T82">XCMPLX_1:82</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="rusub_5.html#E2:56_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xcmplx_1.html#T51">XCMPLX_1:51</a></i>
;<br/><a NAME="E6:56_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="rusub_5.html#E4:56"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E7:56_1_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="bhsp_2.html#D5">BHSP_2:def 5</a></i>;<br/><a NAME="E8:56_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p4"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="rusub_5.html#E4:56"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rusub_5.html#E5:56_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="bhsp_1.html#T32">BHSP_1:32</a></i>;<br/><a NAME="E9:56_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <span class="p3"><a href="bhsp_1.html#K3">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="bhsp_1.html#K3">.||</a></span></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><b>hence </b><a NAME="E10:56_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="rusub_5.html#E7:56_1_1_2"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:56"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>


</div>
