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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">  for <font color="Olive" title="b1">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <font color="Olive" title="b2">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b1">f</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Olive" title="b1">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/>





<a NAME="E1:3"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E2:3"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D2" title="COMPLEX1:def.2">COMPLEX1:def 2</a></span>;<br/>
<a NAME="E3:3"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D3" title="COMSEQ_3:def.3">COMSEQ_3:def 3</a></span>;<br/>
<a NAME="E4:3"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#D1" title="COMPLEX1:def.1">COMPLEX1:def 1</a></span>;<br/>
<a NAME="E5:3"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E6:3"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D3" title="COMSEQ_3:def.3">COMSEQ_3:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><a NAME="E1:3_1"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c3">r</font></span>)</span> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T9" title="COMPLEX1:th.9">COMPLEX1:9</a></span>;<br/><span class="kw">assume </span><a NAME="E2:3_1"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">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"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><a NAME="E3:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="comseq_3.html#D3" title="COMSEQ_3:def.3">COMSEQ_3:def 3</a></span>;<br/><a NAME="E4:3_1"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E6:3"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="ref" href="comseq_3.html#D3" title="COMSEQ_3:def.3">COMSEQ_3:def 3</a></span>;<br/><a NAME="E5:3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E3:3_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</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="E8:3"/>
<font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E6:3"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E3:3"><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"> <font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span><br/>

<a NAME="E9:3"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E10:3"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></span>;<br/>
<a NAME="E11:3"/><span class="lab"><font color="Green" title="E12">A12</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><a NAME="E1:3_2"/><span class="lab"><font color="Green" title="E13">A13</font></span>: 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c3">r</font></span>)</span> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</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> <font color="Maroon" title="c3">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T9" title="COMPLEX1:th.9">COMPLEX1:9</a></span>;<br/><span class="kw">assume </span><a NAME="E2:3_2"/><span class="lab"><font color="Green" title="E14">A14</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">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"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><a NAME="E3:3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A15</font></span>: 
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:3"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></span>;<br/><a NAME="E4:3_2"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:3"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E11:3"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="txt" href="#E10:3"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></span>;<br/><a NAME="E5:3_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:3"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E10:3"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="txt" href="#E1:3_2"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="txt" href="#E3:3_2"><span class="lab"><font color="Green" title="E15">A15</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</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="E13:3"/>
<font color="Maroon" title="c3">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:3"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E11:3"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="txt" href="#E10:3"><span class="lab"><font color="Green" title="E11">A11</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/>


</div>
