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<span class="kw">let </span><font color="Maroon" title="c1">y</font>, <font color="Maroon" title="c2">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">y</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c1">y</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span> )</span><br/>

<a NAME="E2:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" title="XREAL_1:th.63">XREAL_1:63</a></span>;<br/>
<a NAME="E3:18"/>
 <a href="polyeq_1.html#K4" title="POLYEQ_1:func.4">Polynom</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c1">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/>
<a NAME="E4:18"/><span class="kw">then </span>
( <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> or <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><a href="quin_1.html#K2" title="QUIN_1:func.2">delta</a> (1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="polyeq_1.html#T5" title="POLYEQ_1:th.5">POLYEQ_1:5</a></span>;<br/>
<a NAME="E5:18"/><span class="kw">then </span>
( <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2 or <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default">4 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2 )
 
;<br/>
<a NAME="E6:18"/><span class="kw">then </span>
( <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2</span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2 or <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="square_1.html#T20" title="SQUARE_1:th.20">SQUARE_1:20</a>, <a class="ref" href="square_1.html#T29" title="SQUARE_1:th.29">SQUARE_1:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:18"/>
( <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <font color="Maroon" title="c2">a</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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