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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>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<a NAME="E1:66"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">b</font> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
;<br/>
<b>assume </b><a NAME="E2:66"/><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="square_1.html#K3">^2</a>  &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 )
 ;<br/>

<a NAME="E3:66"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:66"/><b>then </b>
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/>
<a NAME="E5:66"/><b>then </b>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<a NAME="E6:66"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<a NAME="E7:66"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="series_5.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:66"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="series_5.html#E4"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="prepower.html#T18">PREPOWER:18</a></i>;<br/>
<a NAME="E9:66"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="series_5.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E10:66"/><b>then </b>
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#T95">SQUARE_1:95</a></i>;<br/>
<a NAME="E11:66"/><b>then </b>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<a NAME="E12:66"/><b>then </b>
<font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<a NAME="E13:66"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="series_5.html#E5"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="prepower.html#T18">PREPOWER:18</a></i>;<br/>
<a NAME="E14:66"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">a</font> <a href="newton.html#K2">|^</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="newton.html#K2">|^</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="series_5.html#E2"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="newton.html#T100">NEWTON:100</a></i>;<br/>
<a NAME="E15:66"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="series_5.html#E1:66"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E16:66"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/>
<a NAME="E17:66"/>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="series_5.html#E2:66"><i><font color="Green">E37</font></i></a>, <a class="txt" href="series_5.html#E6"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E18:66"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/>
<a NAME="E19:66"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="series_5.html#E7"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E20:66"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <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="newton.html#K2">|^</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E21:66"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <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="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T100">NEWTON:100</a></i>;<br/>
<a NAME="E22:66"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="series_5.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="newton.html#T100">NEWTON:100</a></i>;<br/>
<a NAME="E23:66"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="newton.html#K2">|^</a> 2</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T100">NEWTON:100</a></i>;<br/>
<b>hence </b><a NAME="E24:66"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/>


</div>
