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<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K31">compreal</a>  <a href="finseqop.html#K5">*</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:16_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E3:16_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E4:16_1_1"/><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="binop_2.html#K31">compreal</a> 
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:16_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="binop_2.html#K31">compreal</a> 
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:16_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E7:16_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="finseqop.html#K5">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E1"><i><font color="Green">Lemma20</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E8:16_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="finseqop.html#K5">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">fk</font> = <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>, <font color="Maroon">Fk</font> = <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>thus </b><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> = 
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">Fk</font></span>)</span>

<br/>.= 
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D7">BINOP_2:def 7</a></i>
<br/>.= 
 <a href="binop_2.html#K7">-</a> <font color="Maroon">fk</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="binop_2.html#K31">compreal</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D7">BINOP_2:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K31">compreal</a>  <a href="finseqop.html#K5">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:16_1_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K31">compreal</a>  <a href="finseqop.html#K5">*</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


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