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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> implies  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c2">b</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font>
 <span class="kw">and </span><br/><a NAME="E2:27"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c2">b</font></span>)</span></span><br/>

<a NAME="E3:27"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="square_1.html#D2" target="_self" onmouseover="rs('square_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E4:27"/>
<span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="square_1.html#D2" target="_self" onmouseover="rs('square_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E5:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="xcmplx_1.html#T76" onmouseover="rs('xcmplx_1/T76')" onmouseout="rh()">XCMPLX_1:76</a></span>;<br/>
<a NAME="E6:27"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="square_1.html#D2" target="_self" onmouseover="rs('square_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E7:27"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="square_1.html#D2" target="_self" onmouseover="rs('square_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:27"/>
 <a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="square_1.html#D2" target="_self" onmouseover="rs('square_1/D2')" onmouseout="rh()">Def2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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