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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/>



<b>assume </b><a NAME="E1:19"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:19"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="polyeq_1.html#K4">Polynom</a> 1,0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E3:19"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="polyeq_3.html#E1:19"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E4:19"/>
 <a href="quin_1.html#K2">delta</a> 1,0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 4 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E5:19"/><b>then </b>
 <a href="quin_1.html#K2">delta</a> 1,0,<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T121">REAL_2:121</a>, <a class="txt" href="polyeq_3.html#E3:19"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E6:19"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2">delta</a> 1,0,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 0</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2">delta</a> 1,0,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="polyeq_3.html#E2:19"><i><font color="Green">E8</font></i></a>, <a class="ref" href="polyeq_1.html#T5">POLYEQ_1:5</a></i>;<br/>
<a NAME="E7:19"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 
;<br/>
<a NAME="E8:19"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="square_1.html#T85">SQUARE_1:85</a>, <a class="ref" href="square_1.html#T97">SQUARE_1:97</a>, <a class="txt" href="polyeq_3.html#E3:19"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:19"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/>


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