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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <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">i</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c2">n</font> implies (  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</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><br/>



<span class="kw">assume </span><a NAME="E1:23"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c2">n</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="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</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><br/>

<a NAME="E2:23"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c2">n</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">by</span> <span class="lab"><a class="ref" href="power.html#T34" onmouseover="rs('power/T34')" onmouseout="rh()">POWER:34</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:23_1"/>
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><span class="kw">suppose </span></a><a NAME="E1:23_1_1"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> (  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</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><br/><div class="add"><span class="kw">hence </span><a NAME="E2:23_1_1"/>
(  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</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">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_2"><span class="kw">suppose </span></a><a NAME="E1:23_1_2"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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"> (  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</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><br/><div class="add"><a NAME="E2:23_1_2"/><span class="kw">then </span>
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:23_1_2"/>
(  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</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">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E1:23_1_2"><span class="lab"><font color="Green" title="E22">A3</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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