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<span class="kw">let </span><font color="Maroon" title="c1">a</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b1">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  holds <br/>( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Olive" title="b1">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c2">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  implies ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22"/><span class="lab"><font color="Green" title="E12">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="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <span class="kw">and </span><br/><a NAME="E2:22"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"> (  not <font color="Maroon" title="c2">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  or ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:22"/>
<font color="Maroon" title="c2">n</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )</span><br/>

<a NAME="E4:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T9" onmouseover="rs('abian/T9')" onmouseout="rh()">ABIAN:9</a></span>;<br/>
<a NAME="E5:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A4</font></span>: 
( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="abian.html#T12" onmouseover="rs('abian/T12')" onmouseout="rh()">ABIAN:12</a>, <a class="ref" href="xreal_1.html#T25" onmouseover="rs('xreal_1/T25')" onmouseout="rh()">XREAL_1:25</a></span>;<br/>
<a NAME="E6:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="power.html#T20" target="_self" onmouseover="rs('power/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E7:22"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="power.html#T20" target="_self" onmouseover="rs('power/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E8:22"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<a NAME="E9:22"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="xreal_1.html#T26" onmouseover="rs('xreal_1/T26')" onmouseout="rh()">XREAL_1:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:22"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="power.html#T10" target="_self" onmouseover="rs('power/T10')" onmouseout="rh()">Th10</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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