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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="hermitan.html#V5" title="HERMITAN:attr.5">diagReR+0valued</a>  <a href="hermitan.html#NM3" title="HERMITAN:NM.3">hermitan-Form</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font> holds   <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">w</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="hermitan.html#V5" title="HERMITAN:attr.5">diagReR+0valued</a>  <a href="hermitan.html#NM3" title="HERMITAN:NM.3">hermitan-Form</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font> holds   <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">w</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">v3</font> = <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">v4</font> = <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">w1</font> = <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">w2</font> = <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c4">w</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">sv</font> =  <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">sw</font> =  <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">svw</font> =  <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span>;<br/>
<a NAME="E1:97"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#D7" target="_self" title="HERMITAN:def.7">Def7</a></span>;<br/>
<span class="lab"><font color="Green" title="E45">A2</font></span>:  <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> = 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bilinear.html#T29" title="BILINEAR:th.29">BILINEAR:29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#T16" title="HAHNBAN1:th.16">HAHNBAN1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#T16" title="HAHNBAN1:th.16">HAHNBAN1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#T16" title="HAHNBAN1:th.16">HAHNBAN1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#D5" target="_self" title="HERMITAN:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T18" target="_self" title="HERMITAN:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E3:97"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#D7" target="_self" title="HERMITAN:def.7">Def7</a></span>;<br/>
<a NAME="E4:97"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<a NAME="E5:97"/><span class="kw">then </span>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hermitan.html#T48" target="_self" title="HERMITAN:th.48">Th48</a>, <a class="ref" href="square_1.html#T94" title="SQUARE_1:th.94">SQUARE_1:94</a></span>;<br/>
<a NAME="E6:97"/><span class="kw">then </span>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>;<br/>
<a NAME="E7:97"/><span class="kw">then </span>
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> &amp; <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="complex1.html#T140" title="COMPLEX1:th.140">COMPLEX1:140</a>, <a class="ref" href="square_1.html#T97" title="SQUARE_1:th.97">SQUARE_1:97</a></span>;<br/>
<a NAME="E8:97"/><span class="kw">then </span>
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:97"/><span class="kw">then </span>
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E10:97"/><span class="kw">then </span>
 <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p4">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p4">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E11:97"/><span class="kw">then </span>
 <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p4">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p4">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p4">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>;<br/>
<a NAME="E12:97"/><span class="kw">then </span>
 <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p4">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p4">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p4">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p4">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:97"/>
 <a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p3">(<span class="default"><a href="hermitan.html#K4" title="HERMITAN:func.4">signnorm</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
