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<b>let </b><font color="Maroon" title="c1">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E29">A1</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E2:26"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="sin_cos7.html#E9"><i><font color="Green" title="E9">Lm4</font></i></a></i>;<br/>
<a NAME="E3:26"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E4:26"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E5:26"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E29">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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