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<b>let </b><font color="Maroon">t</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="E1:53"/><i><font color="Green">E21</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 ;<br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green">E25</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> or <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> or <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="sin_cos7.html#E39"><i><font color="Green">Lemma61</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>suppose </b></a><a NAME="E1:53_1_1"/><i><font color="Green">E27</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font>
 ;<br/><div class="add"><a NAME="E2:53_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#T76">SQUARE_1:76</a></i>;<br/><a NAME="E3:53_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="sin_cos7.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:53_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
1 <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E5:53_1_1"/>
2 <a href="binop_2.html#K24">*</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="sin_cos7.html#T29" target="_self">Th29</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E6:53_1_1"/><b>then </b>
0 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/><b>hence </b><a NAME="E7:53_1_1"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><b>suppose </b></a><a NAME="E1:53_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:53_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="sin_cos7.html#T28" target="_self">Th28</a>, <a class="ref" href="square_1.html#T75">SQUARE_1:75</a></i>;<br/><a NAME="E3:53_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="ref" href="sin_cos7.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:53_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E5:53_1_2"/>
0 <a href="binop_2.html#K24">*</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="sin_cos7.html#T28" target="_self">Th28</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E6:53_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="sin_cos7.html#T30" target="_self">Th30</a>, <a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><b>hence </b><a NAME="E7:53_1_2"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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