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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1"/>
(  not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>suppose </b></a><a NAME="E1:45_1_1"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:45_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T65" target="_self">Th65</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K11">*</a> 0
<b>by </b><i><a class="txt" href="rvsum_1.html#E1:45_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E2:45_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2"><b>suppose </b></a><a NAME="E1:45_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:45_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E1:45_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E1:45_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rvsum_1.html#E2:45_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funcop_1.html#T42">FUNCOP_1:42</a></i>
<br/>.= 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</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="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

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