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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>assume </b><a NAME="E1:4"/><i><font color="Green">E7</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E2:4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E3:4"/><b>then </b>
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E4:4"/><b>then </b><i><font color="Green">E8</font></i>: 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E5:4"/><b>then </b><i><font color="Green">E9</font></i>: 
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E6:4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E7:4"/><b>then </b>
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E8:4"/><b>then </b>
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E9:4"/><b>then </b><i><font color="Green">E10</font></i>: 
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<div><i><font color="Green">E11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_1"><b>case </b></a><a NAME="E1:4_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:4_1_1_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><b>hence </b><a NAME="E3:4_1_1_1"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:4"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jgraph_6.html#E9:4"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xreal_1.html#T133">XREAL_1:133</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_2"><b>case </b></a><a NAME="E1:4_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:4_1_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:4_1_1_2"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:4"><i><font color="Green">E8</font></i></a>, <a class="ref" href="jgraph_6.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E4:4_1_1_2"/>
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jgraph_6.html#E7"><i><font color="Green">Lemma5</font></i></a></i>;<br/><a NAME="E5:4_1_1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span>
 ;<br/><a NAME="E6:4_1_1_2"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/><a NAME="E7:4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:4"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/><b>hence </b><a NAME="E8:4_1_1_2"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:4_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jgraph_6.html#E5:4_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E11:4"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E12:4"/>
( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> &amp;  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E10:4"><i><font color="Green">E11</font></i></a></i>;<br/>


</div>
