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<b>let </b><font color="Maroon" title="c3">F1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<a NAME="E1:59_1_2"/><i><font color="Green" title="E25">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#NK11" title="RVSUM_1:NK.11">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E2:59_1_2"/><i><font color="Green" title="E26">A2</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<a NAME="E3:59_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<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="E4:59_1_2"/><b>then </b><i><font color="Green" title="E27">A3</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E2:59_1_2"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<b>thus </b><a NAME="E5:59_1_2"/>
( <font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#NK11" title="RVSUM_1:NK.11">*</a> <font color="Maroon" title="c1">F</font> implies <font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">F</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1_2_1"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#NK11" title="RVSUM_1:NK.11">*</a> <font color="Maroon" title="c1">F</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:59_1_2_1_1"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">F1</font>
 ;<br/><b>hence </b><font color="Maroon" title="c3">F1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">c</font> = 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E1:59_1_2_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</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">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E23"><i><font color="Green" title="E6">Lm3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="rvsum_1.html#E1:59_1_2"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="rvsum_1.html#E1:59_1_2_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="rvsum_1.html#E1:59_1_2_1_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:59_1_2_1"/>
<font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E1:59_1_2"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="rvsum_1.html#E4:59_1_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="rvsum_1.html#E1:59_1_2_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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

<b>assume </b><a NAME="E6:59_1_2"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">F</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:59_1_2_2"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">F1</font>
 ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#NK11" title="RVSUM_1:NK.11">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font> = 
<font color="Maroon" title="c2">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66" target="_self" title="RVSUM_1:th.66">Th66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</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">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E23"><i><font color="Green" title="E6">Lm3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#K3" title="RVSUM_1:func.3">multreal</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="rvsum_1.html#E6:59_1_2"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="rvsum_1.html#E1:59_1_2_2"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:59_1_2"/>
<font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="rvsum_1.html#NK11" title="RVSUM_1:NK.11">*</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E1:59_1_2"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="rvsum_1.html#E4:59_1_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="rvsum_1.html#E6:59_1_2"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
