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<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">z</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green">E35</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x</font>, <font color="Maroon">y2</font> = <font color="Maroon">y</font>, <font color="Maroon">z2</font> = <font color="Maroon">z</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">x</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/>
<span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T119">RVSUM_1:119</a></i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="euclid_2.html#K1" target="_self">)|</a></span>

;<br/>
<b>hence </b><a NAME="E4:19"/>
<span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></span><a href="euclid_2.html#K1" 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">x</font>,<font color="Maroon">z</font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="euclid_2.html#K1" target="_self">)|</a></span>
 ;<br/>


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