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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E22</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:25"/><i><font color="Green">E23</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E3:25"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T8" target="_self">Th8</a></i>;<br/>
<b>then </b><span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> = 
<span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span>
<b>by </b><i><a class="txt" href="euclid_2.html#E1:25"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euclid_2.html#E2:25"><i><font color="Green">E23</font></i></a>, <a class="ref" href="euclid_2.html#T19" target="_self">Th19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span>
<b>by </b><i><a class="txt" href="euclid_2.html#E1:25"><i><font color="Green">E22</font></i></a>, <a class="ref" href="euclid_2.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span></span>)</span>
<b>by </b><i><a class="txt" href="euclid_2.html#E1:25"><i><font color="Green">E22</font></i></a>, <a class="ref" href="euclid_2.html#T20" target="_self">Th20</a></i>
;<br/>
<b>hence </b><a NAME="E5:25"/>
<span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span></span>)</span>
 ;<br/>


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