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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="real_1.html#NM1">Real</a>;<br/>



<a NAME="E1:10"/>
 <a href="relset_1.html#K4">dom</a> <a href="rvsum_1.html#K1">sqrreal</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:10"/><b>then </b><i><font color="Green">E10</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="rvsum_1.html#K1">sqrreal</a> 
 
;<br/>
<i><font color="Green">E11</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1">sqrreal</a>  <a href="partfun1.html#K1">*</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#D8">RVSUM_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>
<b>by </b><i><a class="txt" href="topreal6.html#E2:10"><i><font color="Green">E10</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>
;<br/>
<a NAME="E4:10"/><i><font color="Green">E12</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>;<br/>
<a NAME="E5:10"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 ;<br/>

<a NAME="E2:10_2"/><i><font color="Green">E14</font></i>: 
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="topreal6.html#E4:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="topreal6.html#E1:10_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1"><b>suppose </b></a><a NAME="E1:10_2_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="txt" href="topreal6.html#E2:10_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="rvsum_1.html#T78">RVSUM_1:78</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="topreal6.html#E1:10_2_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_2"><b>suppose </b></a><a NAME="E1:10_2_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="txt" href="topreal6.html#E2:10_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="rvsum_1.html#T78">RVSUM_1:78</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="topreal6.html#E1:10_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:10"/>
 <a href="rvsum_1.html#K11">sqr</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="topreal6.html#E3:10"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topreal6.html#E4:10"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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