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<div class="add">

<b>let </b><font color="Maroon" title="c3">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#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>thus </b><a NAME="E1:26_1_1"/>
( <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span> implies (  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">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="c3">IT</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1_1_1"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
 ;<br/>

<b>hence </b><i><font color="Green" title="E18">A2</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">IT</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">seq</font>
<b>by </b><i><a class="ref" href="comseq_1.html#D13" target="_self" title="COMSEQ_1:def.13">Def13</a></i>
<br/>.= 
<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/><br/>

<b>let </b><font color="Maroon" title="c4">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="E3:26_1_1_1"/>
<font color="Maroon" title="c2">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">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="E4:26_1_1_1"/>
<font color="Maroon" title="c3">IT</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="comseq_1.html#E1:26_1_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="comseq_1.html#E2:26_1_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="comseq_1.html#D13" target="_self" title="COMSEQ_1:def.13">Def13</a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:26_1_1"/><i><font color="Green" title="E17">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">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="E3:26_1_1"/><i><font color="Green" title="E18">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="c3">IT</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 ;<br/>

<a NAME="E4:26_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">seq</font>
 
<b>by </b><i><a class="txt" href="comseq_1.html#E2:26_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:26_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="c3">IT</font> holds <br/><font color="Maroon" title="c3">IT</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="E1:26_1_1_2"/>
<font color="Maroon" title="c4">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">IT</font>
 ;<br/>

<a NAME="E2:26_1_1_2"/>
<font color="Maroon" title="c2">seq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">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="E3:26_1_1_2"/>
<font color="Maroon" title="c3">IT</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="comseq_1.html#E3:26_1_1"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:26_1_1"/>
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">|.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="comseq_1.html#K9" title="COMSEQ_1:func.9">.|</a></span>
 <b>by </b><i><a class="txt" href="comseq_1.html#E4:26_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="comseq_1.html#D13" target="_self" title="COMSEQ_1:def.13">Def13</a></i>;<br/>


</div>
