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<span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span></span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span></span><br/>





<span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T29" onmouseover="rs('rlvect_1/T29')" onmouseout="rh()">RLVECT_1:29</a></span>
<br/>.= 
<font color="Maroon" title="c6">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">X</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T15" onmouseover="rs('rlvect_1/T15')" onmouseout="rh()">RLVECT_1:15</a></span>
<br/>.= 
<font color="Maroon" title="c6">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T13" onmouseover="rs('rlvect_1/T13')" onmouseout="rh()">RLVECT_1:13</a></span>
;<br/>
<a NAME="E2:74"/><span class="kw">then </span>
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T46" target="_self" onmouseover="rs('csspace/T46')" onmouseout="rh()">Th41</a></span>;<br/>
<a NAME="E3:74"/><span class="kw">then </span>
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/>
<a NAME="E4:74"/><span class="kw">then </span>
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T33" onmouseover="rs('rlvect_1/T33')" onmouseout="rh()">RLVECT_1:33</a></span>;<br/>
<a NAME="E5:74"/><span class="kw">then </span>
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T47" target="_self" onmouseover="rs('csspace/T47')" onmouseout="rh()">Th42</a></span>;<br/>
<a NAME="E6:74"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A1</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<a NAME="E7:74"/>
<span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace.html#T48" target="_self" onmouseover="rs('csspace/T48')" onmouseout="rh()">Th43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:74"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="csspace.html#K11" title="CSSPACE:func.11">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="csspace.html#K11" title="CSSPACE:func.11">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="absvalue.html#T5" onmouseover="rs('absvalue/T5')" onmouseout="rh()">ABSVALUE:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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