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<div class="add">

<b>let </b><font color="Maroon" title="c4">IT</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">C</font>, <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>

<b>thus </b><a NAME="E1:13_1_1"/>
( <font color="Maroon" title="c4">IT</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c3">f2</font> implies (  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> &amp; ( for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c4">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c4">IT</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c3">f2</font>
 ;<br/>

<b>hence </b><i><font color="Green" title="E10">A2</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">IT</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D2" target="_self" title="COMSEQ_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">C</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">C</font>

;<br/><br/>

<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">C</font>;<br/>

<a NAME="E3:13_1_1_1"/>
( <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <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="ref" href="finseq_4.html#T22" title="FINSEQ_4:th.22">FINSEQ_4:22</a></i>;<br/>
<b>hence </b><a NAME="E4:13_1_1_1"/>
<font color="Maroon" title="c4">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="comseq_1.html#E1:13_1_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="comseq_1.html#E2:13_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="comseq_1.html#D2" target="_self" title="COMSEQ_1:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:13_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E3:13_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c4">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span>
 ;<br/>

<i><font color="Green" title="E11">A5</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">IT</font> = 
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">C</font>
<b>by </b><i><a class="txt" href="comseq_1.html#E2:13_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</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="c2">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E5:13_1_1"/>
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">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="c4">IT</font> holds <br/><font color="Maroon" title="c4">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">c</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3"><b>proof </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">C</font>;<br/>

<b>assume </b><a NAME="E1:13_1_1_3"/>
<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="c4">IT</font>
 ;<br/>

<a NAME="E2:13_1_1_3"/>
( <font color="Maroon" title="c2">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T22" title="FINSEQ_4:th.22">FINSEQ_4:22</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_1_3"/>
<font color="Maroon" title="c4">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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="comseq_1.html#E3:13_1_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:13_1_1"/>
<font color="Maroon" title="c4">IT</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f1</font> <a href="comseq_1.html#K2" title="COMSEQ_1:func.2">+</a> <font color="Maroon" title="c3">f2</font>
 <b>by </b><i><a class="txt" href="comseq_1.html#E4:13_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="comseq_1.html#D2" target="_self" title="COMSEQ_1:def.2">Def2</a></i>;<br/>


</div>
