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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">h</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E15">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>
 ;<br/>

<a NAME="E2:25"/><i><font color="Green" title="E16">A2</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E3:25"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:25"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E4:25"/><b>then </b><i><font color="Green" title="E17">A3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">h</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E2:25"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:25_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">seq</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" target="_self" title="RFUNCT_2:th.9">Th9</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c3">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c3">n</font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:25"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="rfunct_2.html#T21" target="_self" title="RFUNCT_2:th.21">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:25"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="rfunct_2.html#E1:25_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
<b>by </b><i><a class="txt" href="rfunct_2.html#E4:25"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="rfunct_2.html#T21" target="_self" title="RFUNCT_2:th.21">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c3">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#D8" title="SEQ_1:def.8">SEQ_1:def 8</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:25"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c1">seq</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


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