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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green">E16</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:26"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="seqm_3.html#D10">SEQM_3:def 10</a></i>;<br/>
<a NAME="E3:26"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E4:26"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:26"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:26"><i><font color="Green">E16</font></i></a>, <a class="ref" href="rfunct_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="rfunct_2.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seqm_3.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_2.html#E4:26"><i><font color="Green">E17</font></i></a>, <a class="ref" href="rfunct_2.html#D1" target="_self">Def1</a></i>
;<br/>


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