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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">p3</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>







<b>consider </b><font color="Maroon">f1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">f3</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:38"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f3</font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">f2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">f4</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:38"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">f4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">f2</font>,<font color="Maroon">f4</font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">f5</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">f6</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:38"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">f5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">f6</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp; <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">p2</font></span>)</span>,<font color="Maroon">p3</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">f5</font>,<font color="Maroon">f6</font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">f1</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E8:38"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T28">EUCLID:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h2</font> = <font color="Maroon">f2</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E10:38"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T28">EUCLID:28</a></i>;<br/>
<a NAME="E11:38"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T1" target="_self">Th1</a>, <a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E61</font></i>: <font color="Maroon">p1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">p2</font> = 
<font color="Maroon">h1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">h2</font>
<b>by </b><i>, , <a class="ref" href="euclid_2.html#T1" target="_self">Th1</a>, <a class="txt" href="euclid_2.html#E2:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="euclid.html#D10">EUCLID:def 10</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">f2</font>

;<br/>
<a NAME="E13:38"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f3</font> )
 
<b>by </b><i>, <a class="txt" href="euclid_2.html#E2:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="euclid.html#T28">EUCLID:28</a></i>;<br/>
<b>hence </b><a NAME="E14:38"/>
<span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">p2</font></span>)</span>,<font color="Maroon">p3</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p3</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p3</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span>
 <b>by </b><i>, , , <a class="txt" href="euclid_2.html#E4:38"><i><font color="Green">E36</font></i></a>, </i>;<br/>


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