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<b>let </b><font color="Maroon">a</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:37"/>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <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"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos2.html#T1">SIN_COS2:1</a></i>;<br/>
<a NAME="E3:37"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <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"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/>
<b>set </b><font color="Maroon">o</font> = <font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">p</font> = <font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">c</font>;<br/>
<a NAME="E4:37"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<a NAME="E5:37"/>
( <font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<a NAME="E6:37"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <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"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos2.html#T1">SIN_COS2:1</a></i>;<br/>
<a NAME="E7:37"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <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"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T32">POWER:32</a></i>;<br/>
<a NAME="E8:37"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="asympt_1.html#T89">ASYMPT_1:89</a></i>;<br/>
<a NAME="E9:37"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T38">POWER:38</a></i>;<br/>
<a NAME="E10:37"/>
<font color="Maroon">a</font> <a href="prepower.html#K12">#R</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> <a href="prepower.html#K12">#R</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="prepower.html#T96">PREPOWER:96</a></i>;<br/>
<a NAME="E11:37"/><b>then </b>
<font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> <a href="prepower.html#K12">#R</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/>
<a NAME="E12:37"/><b>then </b>
<font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#D2">POWER:def 2</a></i>;<br/>
<a NAME="E13:37"/><b>then </b>
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E14:37"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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