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<b>let </b><font color="Maroon" title="c4">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font> holds   <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span></span><br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span></span><br/>





<a NAME="E1:40"/><i><font color="Green" title="E25">A1</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T38" target="_self" title="BHSP_1:th.38">Th38</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">x</font> = 
<font color="Maroon" title="c6">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43" title="RLVECT_1:th.43">RLVECT_1:43</a></i>
<br/>.= 
<font color="Maroon" title="c6">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">X</font>)
<b>by </b><i><a class="ref" href="rlvect_1.html#T28" title="RLVECT_1:th.28">RLVECT_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>
;<br/>
<a NAME="E3:40"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T36" target="_self" title="BHSP_1:th.36">Th36</a></i>;<br/>
<a NAME="E4:40"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E5:40"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</a></i>;<br/>
<a NAME="E6:40"/><b>then </b>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T37" target="_self" title="BHSP_1:th.37">Th37</a></i>;<br/>
<a NAME="E7:40"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E8:40"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span>
 <b>by </b><i><a class="txt" href="bhsp_1.html#E1:40"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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