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<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> 3</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> 3</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">f2</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="E2:30"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="euclid_2.html#K2">)|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="euclid_2.html#K1">)|</a></span> )
 <b>by </b><i><a class="ref" href="euclid_2.html#D2">EUCLID_2:def 2</a></i>;<br/>
<i><font color="Green">E35</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K2" target="_self">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K3" target="_self">`3</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
3
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<i><font color="Green">E37</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K2" target="_self">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K3" target="_self">`3</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
3
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="euclid_2.html#K2">)|</a></span> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">f1</font>,<font color="Maroon">f2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid_2.html#D1">EUCLID_2:def 1</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 3</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 3</span>)</span></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 2</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 3</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T108">RVSUM_1:108</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 2</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 3</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 3</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 2</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 3</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 3</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K2" target="_self">`2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 2</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 3</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 3</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K2" target="_self">`2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K2" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> 3</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 3</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K2" target="_self">`2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K2" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K3" target="_self">`3</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> 3</span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E6:30"/>
<span class="p1"><a href="euclid_2.html#K2">|(</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="euclid_2.html#K2">)|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K1" target="_self">`1</a> </span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K2" target="_self">`2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K2" target="_self">`2</a> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="euclid_5.html#K3" target="_self">`3</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="euclid_5.html#K3" target="_self">`3</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div>
