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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span><br/>

<a NAME="E1:18"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></span>;<br/>
<a NAME="E2:18"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D11" title="VALUED_1:def.11">VALUED_1:def 11</a></span>;<br/>
<a NAME="E3:18"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:18_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T52" title="COMPLEX1:th.52">COMPLEX1:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T18" title="VALUED_1:th.18">VALUED_1:18</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:18"/>
 <a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <span class="p1">(<span class="default"><a href="valued_1.html#K30" title="VALUED_1:func.30">-</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="valued_1.html#NK57" title="VALUED_1:NK.57">abs</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" 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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