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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:29"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self">Def4</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K1" target="_self">Euclid_add</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="real_ns1.html#E1:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="real_ns1.html#D6" target="_self">Def6</a></i>
;<br/>

<b>thus </b> <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="real_ns1.html#E1:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="real_ns1.html#D6" target="_self">Def6</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K2" target="_self">Euclid_mult</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="real_ns1.html#E1:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="real_ns1.html#D6" target="_self">Def6</a></i>
;<br/>


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