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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">  for <font color="Olive" title="b1">p</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>  st <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Olive" title="b1">p</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</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"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font></span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:68"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:68"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font></span>)</span></span><br/>

<a NAME="E3:68"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="prepower.html#T39" target="_self" onmouseover="rs('prepower/T39')" onmouseout="rh()">Th39</a></span>;<br/>
<a NAME="E4:68"/><span class="lab"><font color="Green" title="E60">A4</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="prepower.html#T39" target="_self" onmouseover="rs('prepower/T39')" onmouseout="rh()">Th39</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font> = 
<span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="prepower.html#K2" title="PREPOWER:func.2">-Root</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K3" title="PREPOWER:func.3">#Z</a> <span class="p3">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T40" target="_self" onmouseover="rs('prepower/T40')" onmouseout="rh()">Th40</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prepower.html#K4" title="PREPOWER:func.4">#Q</a> <font color="Maroon" title="c3">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="ref" href="prepower.html#T22" target="_self" onmouseover="rs('prepower/T22')" onmouseout="rh()">Th22</a>, <a class="ref" href="rat_1.html#T11" onmouseover="rs('rat_1/T11')" onmouseout="rh()">RAT_1:11</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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