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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">e1</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> , <font color="Maroon">e2</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f1</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#K3">F_Real</a> , <font color="Maroon">f2</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <a href="vectsp_1.html#K3">F_Real</a> ;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> )
 ;<br/>

<i><font color="Green">E48</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K12">mlt</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12">mlt</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E3:43"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K12">mlt</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">e1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>, <font color="Maroon">a2</font> = <font color="Maroon">e2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="vectsp_1.html#K3">F_Real</a>  <b>by </b><i><a class="ref" href="vectsp_1.html#D15">VECTSP_1:def 15</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">e2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T86">RVSUM_1:86</a></i>
<br/>.= 
<font color="Maroon">a1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a2</font>
<b>by </b><i><a class="ref" href="matrixr1.html#T2">MATRIXR1:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K12">mlt</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , <a class="ref" href="fvsum_1.html#T73">FVSUM_1:73</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:43"/>
 <a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K12">mlt</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


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