<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">M</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> ;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<i><font color="Green" title="E14">A1</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="ref" href="vfunct_2.html#D6" target="_self" title="VFUNCT_2:def.6">Def6</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vfunct_2.html#D6" target="_self" title="VFUNCT_2:def.6">Def6</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/><b>assume </b><a NAME="E1:33_2"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>
 ;<br/><a NAME="E2:33_2"/><b>then </b>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E3:33_2"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:33_2"/><b>then </b>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="ref" href="vfunct_2.html#D6" target="_self" title="VFUNCT_2:def.6">Def6</a></i>;<br/><a NAME="E5:33_2"/><b>then </b>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="vfunct_2.html#E3:33_2"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:33_2"/><b>then </b><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E7:33_2"/><b>then </b><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_2.html#D6" target="_self" title="VFUNCT_2:def.6">Def6</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">c</font> = 
<span class="p1">(<span class="default"><a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="vfunct_2.html#E1:33_2"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">c</font></span>)</span>
<b>by </b><i><a class="txt" href="vfunct_2.html#E3:33_2"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="vfunct_2.html#D6" target="_self" title="VFUNCT_2:def.6">Def6</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">c</font></span>)</span>
<b>by </b><i><a class="txt" href="vfunct_2.html#E6:33_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="vfunct_2.html#E7:33_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="vfunct_2.html#D6" target="_self" title="VFUNCT_2:def.6">Def6</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:33"/>
<span class="p1">(<span class="default"><a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vfunct_2.html#K6" title="VFUNCT_2:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="vfunct_2.html#E1:33"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="partfun2.html#T3" title="PARTFUN2:th.3">PARTFUN2:3</a></i>;<br/>

<i><font color="Green" title="E15">A6</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2"><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">.||</a></span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="ref" href="vfunct_2.html#D5" target="_self" title="VFUNCT_2:def.5">Def5</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span></span><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">.||</a></span>
<b>by </b><i><a class="ref" href="vfunct_2.html#D5" target="_self" title="VFUNCT_2:def.5">Def5</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/><b>assume </b><a NAME="E1:33_4"/><i><font color="Green" title="E16">A7</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>
 ;<br/><a NAME="E2:33_4"/><b>then </b>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2"><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">.||</a></span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E3:33_4"/><b>then </b><i><font color="Green" title="E17">A8</font></i>: 
( <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">.||</a></span> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:33_4"/><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="vfunct_2.html#E4:33"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="vfunct_2.html#E1:33_4"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="vfunct_2.html#D5" target="_self" title="VFUNCT_2:def.5">Def5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2"><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">c</font> = 
<span class="p1"><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="vfunct_2.html#E1:33_4"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">c</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="txt" href="vfunct_2.html#E3:33_4"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="vfunct_2.html#D5" target="_self" title="VFUNCT_2:def.5">Def5</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">c</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span>
<b>by </b><i><a class="txt" href="vfunct_2.html#E4:33_4"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>
<br/>.= 
<span class="p1"><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span></span><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="vfunct_2.html#E4:33"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="vfunct_2.html#E1:33_4"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="vfunct_2.html#D5" target="_self" title="VFUNCT_2:def.5">Def5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:33"/>
<span class="p1"><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">||.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">.||</a></span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span></span><a href="vfunct_2.html#K5" title="VFUNCT_2:func.5">.||</a></span>
 <b>by </b><i><a class="txt" href="vfunct_2.html#E4:33"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


</div>
