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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.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  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <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;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font></span>)</span>)</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font></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"> (  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <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;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <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="E2:23"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font></span><br/>

<a NAME="E3:23"/>
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</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="newton.html#T83" onmouseover="rs('newton/T83')" onmouseout="rh()">NEWTON:83</a></span>;<br/>
<a NAME="E4:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 <a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>) <a href="integra4.html#R1" title="INTEGRA4:pred.1">divide_into_equal</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="mesfun14.html#D1" target="_self" onmouseover="rs('mesfun14/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E5:23"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:23"/><span class="kw">then </span>
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p2">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/>
<a NAME="E7:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A5</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p4">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</a></span>;<br/>
<a NAME="E8:23"/>
<span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="integra4.html#D1" onmouseover="rs('integra4/D1')" onmouseout="rh()">INTEGRA4:def 1</a></span>;<br/>
<a NAME="E9:23"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D5" onmouseover="rs('integra1/D5')" onmouseout="rh()">INTEGRA1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:23"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p1">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span>)</span>)</span>,1) <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="#E7:23"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="integra1.html#T4" onmouseover="rs('integra1/T4')" onmouseout="rh()">INTEGRA1:4</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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