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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">f</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> &amp;  <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>  holds <br/> ex <font color="Olive" title="b2">F</font> being   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ex <font color="Olive" title="b3">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Olive" title="b1">f</font>,<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="b4">n</font></span>)</span>)</span>)</span>) &amp; (  for <font color="Olive" title="b5">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> ) ) ) ) &amp; (  for <font color="Olive" title="b4">n</font>, <font color="Olive" title="b5">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">m</font> holds <br/> for <font color="Olive" title="b6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b5">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font> ) &amp; (  for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>( <font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b4">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b4">x</font></span>)</span> &amp;  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font> ) ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b2">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>  &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b3">I</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span>) ) &amp; <font color="Olive" title="b3">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b2">F</font></span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">A</font>,<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"> ( <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp;  <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>  implies  ex <font color="Olive" title="b1">F</font> being   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ex <font color="Olive" title="b2">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<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="b3">n</font></span>)</span>)</span>)</span>) &amp; (  for <font color="Olive" title="b4">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> ) ) ) ) &amp; (  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/> for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font> ) &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>( <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font></span>)</span> &amp;  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font> ) ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>  &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b2">I</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>) ) &amp; <font color="Olive" title="b2">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b1">F</font></span>)</span>) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:29"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:29"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E3:29"/><span class="lab"><font color="Green" title="E27">A3</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> 
 ; <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">  ex <font color="Olive" title="b1">F</font> being   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ex <font color="Olive" title="b2">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<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="b3">n</font></span>)</span>)</span>)</span>) &amp; (  for <font color="Olive" title="b4">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> ) ) ) ) &amp; (  for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/> for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font> ) &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>( <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font></span>)</span> &amp;  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font> ) ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b1">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>  &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b2">I</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>) ) &amp; <font color="Olive" title="b2">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Olive" title="b1">F</font></span>)</span>) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">A1</font> = <font color="Maroon" title="c1">A</font> as    <a href="measur12.html#M3" title="MEASUR12:mode.3">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="measur12.html#T72" onmouseover="rs('measur12/T72')" onmouseout="rh()">MEASUR12:72</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>] <span class="kw">means </span>( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> $2 &amp; $2 <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,$2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<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> $1</span>)</span>)</span>)</span>) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) ) &amp;  ex <font color="Olive" title="b1">K</font> being   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> $1</span>)</span>)</span>)</span> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">K</font> holds <br/>( (  <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> $1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Olive" title="b1">K</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b2">k</font> <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="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) &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> $1</span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies ( ( <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Olive" title="b1">K</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</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> $1</span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">k</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) &amp; ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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> $1</span>)</span>)</span>)</span> implies <font color="Olive" title="b1">K</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><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> $1</span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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> $1</span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">k</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) &amp; ( <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> $1</span>)</span>)</span>)</span> implies <font color="Olive" title="b1">K</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b2">k</font> <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"><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> $1</span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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> $1</span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) ) ) ) ) &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> $2 holds <br/> ex <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b3">k</font> &amp; $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> $1</span>)</span>)</span>)</span>,<font color="Olive" title="b3">k</font>)</span>)</span></span>)</span></span>)</span> ) ) ) );<br/>
<a NAME="E5:29"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   ex <font color="Olive" title="b2">g</font> being    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K3" title="RFUNCT_3:func.3">PFuncs</a> (<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">g</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">  ex <font color="Olive" title="b1">g</font> being    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K3" title="RFUNCT_3:func.3">PFuncs</a> (<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">g</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">K</font> being   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> , <font color="Maroon" title="c6">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><span class="kw"> such that </span><br/><a NAME="E2:29_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">K</font> holds <br/>( (  <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#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c5">K</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <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="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) &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#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies ( ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c5">K</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</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#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) &amp; ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span> implies <font color="Maroon" title="c5">K</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><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#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span> implies <font color="Maroon" title="c5">K</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <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"><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#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) ) ) ) ) &amp; <font color="Maroon" title="c6">g</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">K</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">K</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b2">k</font> &amp; <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span>,<font color="Olive" title="b2">k</font>)</span>)</span></span>)</span></span>)</span> ) ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<font color="Maroon" title="c6">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<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#K11" title="NEWTON:func.11">|^</a> <font color="Maroon" title="c4">n</font></span>)</span>)</span>)</span>) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="mesfun14.html#T20" target="_self" onmouseover="rs('mesfun14/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E3:29_1"/>
<font color="Maroon" title="c6">g</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K3" title="RFUNCT_3:func.3">PFuncs</a> (<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T45" onmouseover="rs('partfun1/T45')" onmouseout="rh()">PARTFUN1:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:29_1"/>
 ex <font color="Olive" title="b1">g</font> being    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of  <a href="rfunct_3.html#K3" title="RFUNCT_3:func.3">PFuncs</a> (<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">g</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1"><span class="lab"><font color="Green" title="E29">A5</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>
<span class="kw">consider </span><font color="Maroon" title="c4">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K3" title="RFUNCT_3:func.3">PFuncs</a> (<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:29"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E28">A4</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">F</font> as   <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>
<a NAME="E9:29"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds   <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">n</font>, <font color="Maroon" title="c7">m</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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span></span><br/>

<a NAME="E1:29_2"/>
( <font color="Maroon" title="c6">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <font color="Maroon" title="c7">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E2:29_2"/><span class="kw">then </span>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c6">n</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">m</font>,<font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">m</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E29">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:29_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c5">F</font> as   <a href="mesfunc8.html#V1" title="MESFUNC8:attr.1">with_the_same_dom</a>  <a href="seqfunc.html#NM1" title="SEQFUNC:NM.1">Functional_Sequence</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc8.html#D2" onmouseover="rs('mesfunc8/D2')" onmouseout="rh()">MESFUNC8:def 2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">F</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">  ex <font color="Olive" title="b1">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<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="b2">n</font></span>)</span>)</span>)</span>) &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> ) ) ) ) &amp; (  for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font> ) &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>( <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font></span>)</span> &amp;  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font> ) ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>  &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>) ) &amp; <font color="Olive" title="b1">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c6">F</font></span>)</span>) )</span><br/>

<a NAME="E11:29"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E29">A6</font></span></a></span>;<br/>
<a NAME="E12:29"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<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>) &amp; (  for <font color="Olive" title="b2">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<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="c7">n</font></span>)</span>)</span>)</span>) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) ) )</span><br/>

<a NAME="E1:29_3"/>
<font color="Maroon" title="c7">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:29_3"/>
( <font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<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="c7">n</font></span>)</span>)</span>)</span>) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E29">A6</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>
<a NAME="E13:29"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font>, <font color="Maroon" title="c8">m</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="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> implies  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_4"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</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">  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span><br/>

<a NAME="E2:29_4"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
( <font color="Maroon" title="c7">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <font color="Maroon" title="c8">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> ( <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:29_4"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">Kn</font> being   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> <span class="kw"> such that </span><br/><a NAME="E5:29_4"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">Kn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c7">n</font></span>)</span>)</span>)</span> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">Kn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">Kn</font> holds <br/>( (  <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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <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="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) &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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies ( ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) &amp; ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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="c7">n</font></span>)</span>)</span>)</span> implies <font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">n</font></span>)</span>)</span>)</span> implies <font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <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"><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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) ) ) ) ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">Kn</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b2">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>)</span>)</span>,<font color="Olive" title="b2">k</font>)</span>)</span></span>)</span></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E2:29_4"><span class="lab"><font color="Green" title="E34">A11</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">Km</font> being   <a href="mesfunc2.html#NM1" title="MESFUNC2:NM.1">Finite_Sep_Sequence</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> <span class="kw"> such that </span><br/><a NAME="E7:29_4"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Km</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c8">m</font></span>)</span>)</span>)</span> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">Km</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Km</font> holds <br/>( (  <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="c8">m</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c11">Km</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <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="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="supinf_2.html#K5" title="SUPINF_2:func.5">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) &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="c8">m</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies ( ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c11">Km</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) &amp; ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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="c8">m</font></span>)</span>)</span>)</span> implies <font color="Maroon" title="c11">Km</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c8">m</font></span>)</span>)</span>)</span> implies <font color="Maroon" title="c11">Km</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b1">k</font> <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"><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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> ) ) ) ) ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Km</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Km</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Olive" title="b2">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span>)</span>)</span>,<font color="Olive" title="b2">k</font>)</span>)</span></span>)</span></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E2:29_4"><span class="lab"><font color="Green" title="E34">A11</font></span></a></span>;<br/>
<a NAME="E8:29_4"/><span class="lab"><font color="Green" title="E38">A15</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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">Kn</font> &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="c8">m</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Km</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/>
<a NAME="E9:29_4"/>
<font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E2:29_4"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E3:29_4"><span class="lab"><font color="Green" title="E35">A12</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">k1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E11:29_4"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k1</font> &amp; <font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">Kn</font> &amp; <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c12">k1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a></span>;<br/>
<a NAME="E12:29_4"/>
<font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E2:29_4"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E3:29_4"><span class="lab"><font color="Green" title="E35">A12</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">k2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E14:29_4"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k2</font> &amp; <font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Km</font> &amp; <font color="Maroon" title="c9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">Km</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c13">k2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a></span>;<br/>
<a NAME="E15:29_4"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
( <font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <font color="Maroon" title="c13">k2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E14:29_4"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E16:29_4"/><span class="lab"><font color="Green" title="E42">A19</font></span>: 
( <font color="Maroon" title="c12">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">Kn</font> &amp; <font color="Maroon" title="c13">k2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Km</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E14:29_4"><span class="lab"><font color="Green" title="E40">A17</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="E17:29_4"/><span class="kw">then </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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="integra1.html#T8" onmouseover="rs('integra1/T8')" onmouseout="rh()">INTEGRA1:8</a></span>;<br/>
<a NAME="E18:29_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A20</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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>;<br/>
<a NAME="E19:29_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A21</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<a NAME="E20:29_4"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
;<br/>
<a NAME="E21:29_4"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E18:29_4"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="rfunct_1.html#T74" onmouseover="rs('rfunct_1/T74')" onmouseout="rh()">RFUNCT_1:74</a></span>;<br/>
<a NAME="E22:29_4"/><span class="kw">then </span>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">real-bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T15" onmouseover="rs('integra1/T15')" onmouseout="rh()">INTEGRA1:15</a></span>;<br/>
<a NAME="E23:29_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A22</font></span>: 
(  not  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:29_4"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="relat_1.html#T42" onmouseover="rs('relat_1/T42')" onmouseout="rh()">RELAT_1:42</a></span>;<br/>
<a NAME="E24:29_4"/>
 <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="integra1.html#T8" onmouseover="rs('integra1/T8')" onmouseout="rh()">INTEGRA1:8</a></span>;<br/>
<a NAME="E25:29_4"/><span class="kw">then </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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>;<br/>
<a NAME="E26:29_4"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<a NAME="E27:29_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A23</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T42" onmouseover="rs('relat_1/T42')" onmouseout="rh()">RELAT_1:42</a></span>;<br/>
<a NAME="E28:29_4"/><span class="lab"><font color="Green" title="E47">A24</font></span>: 
( 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">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>  &amp; 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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="E29:29_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A25</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="c7">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="c7">n</font> &amp;  <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="c8">m</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="c8">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="mesfun14.html#D1" target="_self" onmouseover="rs('mesfun14/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E30:29_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A26</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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> &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="c8">m</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra4.html#D1" onmouseover="rs('integra4/D1')" onmouseout="rh()">INTEGRA4:def 1</a></span>;<br/>
<a NAME="E31:29_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A27</font></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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font></span>)</span> &amp; <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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">k2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:29_4"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E14:29_4"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="integra4.html#D1" onmouseover="rs('integra4/D1')" onmouseout="rh()">INTEGRA4:def 1</a></span>;<br/>
<a NAME="E32:29_4"/><span class="lab"><font color="Green" title="E51">A28</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">n</font> <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="#E1:29_4"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a></span>;<br/>
<a NAME="E33:29_4"/><span class="lab"><font color="Green" title="E52">A29</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E34:29_4"/>
( 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:29_4"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/>
<a NAME="E35:29_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A30</font></span>: 
( <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T48" onmouseover="rs('xreal_1/T48')" onmouseout="rh()">XREAL_1:48</a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E54">A31</font></span>: <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T8" onmouseover="rs('newton/T8')" onmouseout="rh()">NEWTON:8</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:29_4"><span class="lab"><font color="Green" title="E51">A28</font></span></a>, <a class="ref" href="nat_d.html#T72" onmouseover="rs('nat_d/T72')" onmouseout="rh()">NAT_D:72</a></span>
;<br/>
<a NAME="E37:29_4"/><span class="lab"><font color="Green" title="E55">A32</font></span>: 
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" onmouseover="rs('nat_d/T35')" onmouseout="rh()">NAT_D:35</a>, <a class="ref" href="prepower.html#T93" onmouseover="rs('prepower/T93')" onmouseout="rh()">PREPOWER:93</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E56">A33</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_4_2"><span class="kw">now </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"> ( 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">k1</font> implies <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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</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="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_4_2"/>
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">k1</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"> <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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</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="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><br/><a NAME="E2:29_4_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:29_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A34</font></span>: 
<font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" onmouseover="rs('nat_d/T36')" onmouseout="rh()">NAT_D:36</a></span>;<br/><a NAME="E4:29_4_2"/>
<font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" onmouseover="rs('nat_d/T35')" onmouseout="rh()">NAT_D:35</a></span>;<br/><a NAME="E5:29_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">Kn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:29_4_2"/>
<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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</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="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:29_4"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="txt" href="#E30:29_4"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="txt" href="#E3:29_4_2"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="integra4.html#D1" onmouseover="rs('integra4/D1')" onmouseout="rh()">INTEGRA4:def 1</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="lab"><font color="Green" title="E57">A35</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_4_3"><span class="kw">now </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"> ( 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">k2</font> implies <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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_4_3"/>
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">k2</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"> <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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span><br/><a NAME="E2:29_4_3"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_4"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:29_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A36</font></span>: 
<font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" onmouseover="rs('nat_d/T36')" onmouseout="rh()">NAT_D:36</a></span>;<br/><a NAME="E4:29_4_3"/>
<font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" onmouseover="rs('nat_d/T35')" onmouseout="rh()">NAT_D:35</a></span>;<br/><a NAME="E5:29_4_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Km</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_4"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:29_4_3"/>
<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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:29_4"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="txt" href="#E30:29_4"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="txt" href="#E3:29_4_3"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="integra4.html#D1" onmouseover="rs('integra4/D1')" onmouseout="rh()">INTEGRA4:def 1</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>
<a NAME="E40:29_4"/><span class="lab"><font color="Green" title="E58">A37</font></span>: 
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span> <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="E41:29_4"/>
( <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font> &amp; <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="mesfun14.html#E23"><span class="lab"><font color="Green" title="E19">Lm5</font></span></a></span>;<br/>
<a NAME="E42:29_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A38</font></span>: 
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:29_4"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="ref" href="xcmplx_1.html#T89" onmouseover="rs('xcmplx_1/T89')" onmouseout="rh()">XCMPLX_1:89</a></span>;<br/>
<a NAME="E43:29_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A39</font></span>: 
<span class="p1">(<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> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T81" onmouseover="rs('xcmplx_1/T81')" onmouseout="rh()">XCMPLX_1:81</a></span>;<br/>
<a NAME="E44:29_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A40</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E45:29_4"/>
 <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_4_4"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_4_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="E1:29_4_4_1"/>
(  <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="c8">m</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <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="c8">m</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_1"/>
 <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="c8">m</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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span><br/><div class="add"><a NAME="E2:29_4_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A41</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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:29_4"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="txt" href="#E1:29_4"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="prepower.html#T93" onmouseover="rs('prepower/T93')" onmouseout="rh()">PREPOWER:93</a>, <a class="ref" href="nat_1.html#T25" onmouseover="rs('nat_1/T25')" onmouseout="rh()">NAT_1:25</a></span>;<br/><a NAME="E3:29_4_4_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">Kn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/><a NAME="E4:29_4_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:29_4_4_1_1"/><span class="kw">then </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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>) <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="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E2:29_4_4_1_1"><span class="lab"><font color="Green" title="E62">A41</font></span></a>, <a class="txt" href="mesfun14.html#E22"><span class="lab"><font color="Green" title="E18">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:29_4_4_1_1"/>
 <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="integra1.html#T8" onmouseover="rs('integra1/T8')" onmouseout="rh()">INTEGRA1:8</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="29_4_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2"/><span class="lab"><font color="Green" title="E62">A42</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="c8">m</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_4_1_2_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="E1:29_4_4_1_2_1"/>
(  <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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or (  <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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c13">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or (  <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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c13">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c8">m</font></span>)</span>)</span>)</span> ) or (  <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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c13">k2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c13">k2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="c8">m</font></span>)</span>)</span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_4_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2_1_1"/><span class="lab"><font color="Green" title="E63">A43</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="c7">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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span><br/><div class="add"><a NAME="E2:29_4_4_1_2_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">Kn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/><a NAME="E3:29_4_4_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E4:29_4_4_1_2_1_1"/><span class="kw">then </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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>) <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="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_1"><span class="lab"><font color="Green" title="E63">A43</font></span></a>, <a class="txt" href="mesfun14.html#E22"><span class="lab"><font color="Green" title="E18">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:29_4_4_1_2_1_1"/>
 <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="integra1.html#T8" onmouseover="rs('integra1/T8')" onmouseout="rh()">INTEGRA1:8</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="29_4_4_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2_1_2"/><span class="lab"><font color="Green" title="E63">A44</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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c13">k2</font> <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span><br/><div class="add"><a NAME="E2:29_4_4_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">Km</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c13">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2"><span class="lab"><font color="Green" title="E62">A42</font></span></a></span>;<br/><a NAME="E3:29_4_4_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A45</font></span>: 
(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</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="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_4"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="xxreal_1.html#T3" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">XXREAL_1:3</a></span>;<br/><a NAME="E4:29_4_4_1_2_1_2"/>
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="newton.html#T89" onmouseover="rs('newton/T89')" onmouseout="rh()">NEWTON:89</a></span>;<br/><a NAME="E5:29_4_4_1_2_1_2"/><span class="kw">then </span>
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:29_4"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</a></span>;<br/><a NAME="E6:29_4_4_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A46</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E28:29_4"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="ref" href="xreal_1.html#T118" onmouseover="rs('xreal_1/T118')" onmouseout="rh()">XREAL_1:118</a></span>;<br/><a NAME="E7:29_4_4_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E8:29_4_4_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_4_4_1_2_1_2"><span class="lab"><font color="Green" title="E65">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E9:29_4_4_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A47</font></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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4_4_1_2_1_2"><span class="lab"><font color="Green" title="E63">A44</font></span></a>, <a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_4_4_1_2_1_2_1"><span class="kw">now </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">  not <font color="Maroon" title="c12">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_4_4_1_2_1_2_1"/><span class="lab"><font color="Green" title="E67">A48</font></span>: 
<font color="Maroon" title="c12">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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"> contradiction</span><br/><a NAME="E2:29_4_4_1_2_1_2_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:29_4_4_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" onmouseover="rs('nat_d/T36')" onmouseout="rh()">NAT_D:36</a></span>;<br/><a NAME="E4:29_4_4_1_2_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E5:29_4_4_1_2_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_4_4_1_2_1_2"><span class="lab"><font color="Green" title="E65">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:29_4_4_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A49</font></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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4_4_1_2_1_2"><span class="lab"><font color="Green" title="E63">A44</font></span></a>, <a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_2_1"><span class="lab"><font color="Green" title="E67">A48</font></span></a>, <a class="txt" href="#E38:29_4"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_4_1_2_1_2_1_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="E1:29_4_4_1_2_1_2_1_1"/>
( ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k1</font> &amp; <font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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="c7">n</font></span>)</span>)</span>)</span> ) or <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">n</font></span>)</span>)</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4_4_1_2_1_2_1"><span class="lab"><font color="Green" title="E67">A48</font></span></a>, <a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E8:29_4"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_4_1_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2_1_2_1_1_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k1</font> &amp; <font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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="c7">n</font></span>)</span>)</span>)</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"> contradiction</span><br/><div class="add"><a NAME="E2:29_4_4_1_2_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a></span>;<br/><a NAME="E3:29_4_4_1_2_1_2_1_1_1"/><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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_1.html#T3" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">XXREAL_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:29_4_4_1_2_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_4_4_1_2_1_2"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="txt" href="#E6:29_4_4_1_2_1_2_1"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="29_4_4_1_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">n</font></span>)</span>)</span>)</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"> contradiction</span><br/><div class="add"><a NAME="E2:29_4_4_1_2_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c12">k1</font> <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"><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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font></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="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_2"><span class="lab"><font color="Green" title="E63">A44</font></span></a></span>;<br/><a NAME="E3:29_4_4_1_2_1_2_1_1_2"/><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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:29_4_4_1_2_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_4_4_1_2_1_2"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="txt" href="#E6:29_4_4_1_2_1_2_1"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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></div><span class="kw">end;</span></div><a NAME="E11:29_4_4_1_2_1_2"/><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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E12:29_4_4_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A50</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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>) <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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font></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="E13:29_4_4_1_2_1_2"/>
(  <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4_4_1_2_1_2"><span class="lab"><font color="Green" title="E63">A44</font></span></a>, <a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E14:29_4_4_1_2_1_2"/><span class="kw">then </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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font></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/><span class="kw">hence </span><a NAME="E15:29_4_4_1_2_1_2"/>
 <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_4_4_1_2_1_2"><span class="lab"><font color="Green" title="E66">A47</font></span></a>, <a class="txt" href="#E12:29_4_4_1_2_1_2"><span class="lab"><font color="Green" title="E67">A50</font></span></a>, <a class="ref" href="xxreal_1.html#T34" onmouseover="rs('xxreal_1/T34')" onmouseout="rh()">XXREAL_1:34</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="29_4_4_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2_1_3"/><span class="lab"><font color="Green" title="E63">A51</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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c13">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c8">m</font></span>)</span>)</span>)</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">  <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span><br/><div class="add"><a NAME="E2:29_4_4_1_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">Km</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c13">k2</font> <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"><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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font></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="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2"><span class="lab"><font color="Green" title="E62">A42</font></span></a></span>;<br/><a NAME="E3:29_4_4_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A52</font></span>: 
( <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">k2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2"><span class="lab"><font color="Green" title="E62">A42</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E39:29_4"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E14:29_4"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E4:29_4_4_1_2_1_3"/><span class="lab"><font color="Green" title="E65">A53</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:29_4"><span class="lab"><font color="Green" title="E59">A38</font></span></a>, <a class="ref" href="xcmplx_1.html#T81" onmouseover="rs('xcmplx_1/T81')" onmouseout="rh()">XCMPLX_1:81</a></span>;<br/><div><span class="lab"><font color="Green" title="E66">A54</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_4_4_1_2_1_3_1"><span class="kw">now </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">  not <font color="Maroon" title="c12">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="c7">n</font></span>)</span>)</span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_4_4_1_2_1_3_1"/><span class="lab"><font color="Green" title="E67">A55</font></span>: 
<font color="Maroon" title="c12">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="c7">n</font></span>)</span>)</span>)</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"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_4_1_2_1_3_1_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="E1:29_4_4_1_2_1_3_1_1"/>
( <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k1</font> &amp; <font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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="c7">n</font></span>)</span>)</span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:29_4"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_3_1"><span class="lab"><font color="Green" title="E67">A55</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_4_1_2_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E68">A56</font></span>: 
<font color="Maroon" title="c12">k1</font> <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"> contradiction</span><br/><div class="add"><a NAME="E2:29_4_4_1_2_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</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="c7">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#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a></span>;<br/><a NAME="E3:29_4_4_1_2_1_3_1_1_1"/><span class="kw">then </span>
(  <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</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="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c7">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="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_1.html#T3" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">XXREAL_1:3</a></span>;<br/><a NAME="E4:29_4_4_1_2_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A57</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4_4_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E68">A56</font></span></a>, <a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E42:29_4"><span class="lab"><font color="Green" title="E59">A38</font></span></a>, <a class="ref" href="xcmplx_1.html#T82" onmouseover="rs('xcmplx_1/T82')" onmouseout="rh()">XCMPLX_1:82</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_4_4_1_2_1_3_1_1_1_1"><span class="kw">now </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">  not 2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_4_4_1_2_1_3_1_1_1_1"/>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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"> contradiction</span><br/><a NAME="E2:29_4_4_1_2_1_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:29_4"><span class="lab"><font color="Green" title="E52">A29</font></span></a>, <a class="ref" href="pepin.html#T30" onmouseover="rs('pepin/T30')" onmouseout="rh()">PEPIN:30</a></span>;<br/><a NAME="E3:29_4_4_1_2_1_3_1_1_1_1"/><span class="kw">then </span>
2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T4" onmouseover="rs('newton/T4')" onmouseout="rh()">NEWTON:4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:29_4_4_1_2_1_3_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E29:29_4"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="ref" href="integra4.html#D1" onmouseover="rs('integra4/D1')" onmouseout="rh()">INTEGRA4:def 1</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><a NAME="E6:29_4_4_1_2_1_3_1_1_1"/><span class="kw">then </span>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:29_4"><span class="lab"><font color="Green" title="E55">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E7:29_4_4_1_2_1_3_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:29_4_4_1_2_1_3_1_1_1"/><span class="kw">then </span>
2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:29_4"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E30:29_4"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E9:29_4_4_1_2_1_3_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E10:29_4_4_1_2_1_3_1_1_1"/><span class="kw">then </span>
<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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E11:29_4_4_1_2_1_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="txt" href="#E4:29_4_4_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E69">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="29_4_4_1_2_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2_1_3_1_1_2"/><span class="lab"><font color="Green" title="E68">A58</font></span>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k1</font> &amp; <font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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="c7">n</font></span>)</span>)</span>)</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"> contradiction</span><br/><div class="add"><a NAME="E2:29_4_4_1_2_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a></span>;<br/><a NAME="E3:29_4_4_1_2_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A59</font></span>: 
( <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</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="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E68">A58</font></span></a>, <a class="txt" href="#E38:29_4"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_1.html#T3" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">XXREAL_1:3</a></span>;<br/><a NAME="E4:29_4_4_1_2_1_3_1_1_2"/>
<font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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="c7">n</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4_4_1_2_1_3_1"><span class="lab"><font color="Green" title="E67">A55</font></span></a>, <a class="txt" href="#E8:29_4"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:29_4_4_1_2_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:29_4"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E6:29_4_4_1_2_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E7:29_4_4_1_2_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A60</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E8:29_4_4_1_2_1_3_1_1_2"/>
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:29_4"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E30:29_4"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="txt" href="#E40:29_4"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/><a NAME="E9:29_4_4_1_2_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:29_4"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E10:29_4_4_1_2_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E65">A53</font></span></a>, <a class="txt" href="#E7:29_4_4_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E70">A60</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E11:29_4_4_1_2_1_3_1_1_2"/><span class="kw">then </span>
<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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E12:29_4_4_1_2_1_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E64">A52</font></span></a>, <a class="txt" href="#E3:29_4_4_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E69">A59</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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></div><span class="kw">end;</span></div><a NAME="E6:29_4_4_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A61</font></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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</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="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> &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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E38:29_4"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E15:29_4"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E7:29_4_4_1_2_1_3"/><span class="lab"><font color="Green" title="E68">A62</font></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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> &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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span> <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">k2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E39:29_4"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E15:29_4"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2"><span class="lab"><font color="Green" title="E62">A42</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E8:29_4_4_1_2_1_3"/>
<span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:29_4"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="txt" href="#E40:29_4"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T10" onmouseover="rs('xreal_1/T10')" onmouseout="rh()">XREAL_1:10</a></span>;<br/><a NAME="E9:29_4_4_1_2_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E66">A54</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E30:29_4"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="txt" href="#E15:29_4"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E10:29_4_4_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A63</font></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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E65">A53</font></span></a>, <a class="txt" href="#E5:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E66">A54</font></span></a>, <a class="txt" href="#E15:29_4"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E30:29_4"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="txt" href="#E6:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E67">A61</font></span></a>, <a class="txt" href="#E7:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E68">A62</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E11:29_4_4_1_2_1_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E66">A54</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E63">A51</font></span></a>, <a class="txt" href="#E30:29_4"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="ref" href="xcmplx_1.html#T82" onmouseover="rs('xcmplx_1/T82')" onmouseout="rh()">XCMPLX_1:82</a></span>;<br/><a NAME="E12:29_4_4_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A64</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:29_4"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="ref" href="xcmplx_1.html#T51" onmouseover="rs('xcmplx_1/T51')" onmouseout="rh()">XCMPLX_1:51</a></span>;<br/><a NAME="E13:29_4_4_1_2_1_3"/>
(  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>) <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> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <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> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span></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/><span class="kw">hence </span><a NAME="E14:29_4_4_1_2_1_3"/>
 <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E67">A61</font></span></a>, <a class="txt" href="#E7:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E68">A62</font></span></a>, <a class="txt" href="#E10:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E69">A63</font></span></a>, <a class="txt" href="#E12:29_4_4_1_2_1_3"><span class="lab"><font color="Green" title="E70">A64</font></span></a>, <a class="ref" href="xxreal_1.html#T34" onmouseover="rs('xxreal_1/T34')" onmouseout="rh()">XXREAL_1:34</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="29_4_4_1_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2_1_4"/><span class="lab"><font color="Green" title="E63">A65</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="c7">n</font></span>)</span>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c13">k2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c13">k2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="c8">m</font></span>)</span>)</span>)</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">  <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span><br/><div class="add"><a NAME="E2:29_4_4_1_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A66</font></span>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">k2</font> &amp; <font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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="c8">m</font></span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_4"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="txt" href="#E8:29_4"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:29_4_4_1_2_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">Km</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c13">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a></span>;<br/><a NAME="E4:29_4_4_1_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A67</font></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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29_4"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="xxreal_1.html#T3" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">XXREAL_1:3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_4_1_2_1_4_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="E1:29_4_4_1_2_1_4_1"/>
( <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k1</font> &amp; <font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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="c7">n</font></span>)</span>)</span>)</span> ) or <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">n</font></span>)</span>)</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E8:29_4"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_4_4_1_2_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2_1_4_1_1"/><span class="lab"><font color="Green" title="E66">A68</font></span>: 
<font color="Maroon" title="c12">k1</font> <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span><br/><div class="add"><a NAME="E2:29_4_4_1_2_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a></span>;<br/><a NAME="E3:29_4_4_1_2_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A69</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_1.html#T3" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">XXREAL_1:3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_4_4_1_2_1_4_1_1_1"><span class="kw">now </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">  not <font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">k2</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_4_4_1_2_1_4_1_1_1"/>
<font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">k2</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"> contradiction</span><br/><a NAME="E2:29_4_4_1_2_1_4_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:29_4_4_1_2_1_4_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T42" onmouseover="rs('nat_d/T42')" onmouseout="rh()">NAT_D:42</a></span>;<br/><a NAME="E4:29_4_4_1_2_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A70</font></span>: 
<font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>;<br/><a NAME="E5:29_4_4_1_2_1_4_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_4_4_1_2_1_4_1_1_1"><span class="lab"><font color="Green" title="E68">A70</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E6:29_4_4_1_2_1_4_1_1_1"/><span class="kw">then </span>
<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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:29_4"><span class="lab"><font color="Green" title="E60">A39</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E7:29_4_4_1_2_1_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E65">A67</font></span></a>, <a class="txt" href="#E39:29_4"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="txt" href="#E3:29_4_4_1_2_1_4_1_1"><span class="lab"><font color="Green" title="E67">A69</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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><a NAME="E5:29_4_4_1_2_1_4_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:29_4"><span class="lab"><font color="Green" title="E61">A40</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E6:29_4_4_1_2_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A71</font></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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E7:29_4_4_1_2_1_4_1_1"/><span class="lab"><font color="Green" title="E69">A72</font></span>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:29_4_4_1_2_1_4_1_1"/><span class="lab"><font color="Green" title="E70">A73</font></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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4_1_1"><span class="lab"><font color="Green" title="E66">A68</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E9:29_4_4_1_2_1_4_1_1"/><span class="lab"><font color="Green" title="E71">A74</font></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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E15:29_4"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E10:29_4_4_1_2_1_4_1_1"/>
(  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>) <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> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <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> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span></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/><span class="kw">hence </span><a NAME="E11:29_4_4_1_2_1_4_1_1"/>
 <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_4_4_1_2_1_4_1_1"><span class="lab"><font color="Green" title="E68">A71</font></span></a>, <a class="txt" href="#E7:29_4_4_1_2_1_4_1_1"><span class="lab"><font color="Green" title="E69">A72</font></span></a>, <a class="txt" href="#E39:29_4"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="txt" href="#E8:29_4_4_1_2_1_4_1_1"><span class="lab"><font color="Green" title="E70">A73</font></span></a>, <a class="txt" href="#E9:29_4_4_1_2_1_4_1_1"><span class="lab"><font color="Green" title="E71">A74</font></span></a>, <a class="ref" href="xxreal_1.html#T34" onmouseover="rs('xxreal_1/T34')" onmouseout="rh()">XXREAL_1:34</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="29_4_4_1_2_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2_1_4_1_2"/><span class="lab"><font color="Green" title="E66">A75</font></span>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k1</font> &amp; <font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <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="c7">n</font></span>)</span>)</span>)</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">  <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span><br/><div class="add"><a NAME="E2:29_4_4_1_2_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font></span>)</span></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a></span>;<br/><a NAME="E3:29_4_4_1_2_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A76</font></span>: 
( <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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</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="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_4_4_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E66">A75</font></span></a>, <a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E38:29_4"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_1.html#T3" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">XXREAL_1:3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_4_4_1_2_1_4_1_2_1"><span class="kw">now </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">  not <font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">k2</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_4_4_1_2_1_4_1_2_1"/>
<font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">k2</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"> contradiction</span><br/><a NAME="E2:29_4_4_1_2_1_4_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:29_4_4_1_2_1_4_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T42" onmouseover="rs('nat_d/T42')" onmouseout="rh()">NAT_D:42</a></span>;<br/><a NAME="E4:29_4_4_1_2_1_4_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>;<br/><a NAME="E5:29_4_4_1_2_1_4_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E6:29_4_4_1_2_1_4_1_2_1"/><span class="kw">then </span>
<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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</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="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:29_4"><span class="lab"><font color="Green" title="E60">A39</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E7:29_4_4_1_2_1_4_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E65">A67</font></span></a>, <a class="txt" href="#E39:29_4"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="txt" href="#E3:29_4_4_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E67">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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><a NAME="E5:29_4_4_1_2_1_4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:29_4"><span class="lab"><font color="Green" title="E61">A40</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E6:29_4_4_1_2_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A77</font></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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E7:29_4_4_1_2_1_4_1_2"/><span class="lab"><font color="Green" title="E69">A78</font></span>: 
<font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E64">A66</font></span></a>, <a class="ref" href="nat_d.html#T36" onmouseover="rs('nat_d/T36')" onmouseout="rh()">NAT_D:36</a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_4_4_1_2_1_4_1_2_2"><span class="kw">now </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">  not <font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_4_4_1_2_1_4_1_2_2"/>
<font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</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"> contradiction</span><br/><a NAME="E2:29_4_4_1_2_1_4_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:29_4_4_1_2_1_4_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_4_4_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E69">A78</font></span></a>, <a class="ref" href="nat_d.html#T43" onmouseover="rs('nat_d/T43')" onmouseout="rh()">NAT_D:43</a></span>;<br/><a NAME="E4:29_4_4_1_2_1_4_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E5:29_4_4_1_2_1_4_1_2_2"/><span class="kw">then </span>
<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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">k2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</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="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:29_4"><span class="lab"><font color="Green" title="E60">A39</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:29_4_4_1_2_1_4_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_4_4_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E67">A76</font></span></a>, <a class="txt" href="#E4:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E65">A67</font></span></a>, <a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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><a NAME="E9:29_4_4_1_2_1_4_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:29_4"><span class="lab"><font color="Green" title="E61">A40</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E10:29_4_4_1_2_1_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A79</font></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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:29_4"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="txt" href="#E39:29_4"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E66">A75</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E11:29_4_4_1_2_1_4_1_2"/><span class="lab"><font color="Green" title="E71">A80</font></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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E15:29_4"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E66">A75</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E12:29_4_4_1_2_1_4_1_2"/><span class="lab"><font color="Green" title="E72">A81</font></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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E15:29_4"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E13:29_4_4_1_2_1_4_1_2"/>
(  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>) <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> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <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> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span></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/><span class="kw">hence </span><a NAME="E14:29_4_4_1_2_1_4_1_2"/>
 <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_4_4_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E68">A77</font></span></a>, <a class="txt" href="#E10:29_4_4_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E70">A79</font></span></a>, <a class="txt" href="#E11:29_4_4_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E71">A80</font></span></a>, <a class="txt" href="#E12:29_4_4_1_2_1_4_1_2"><span class="lab"><font color="Green" title="E72">A81</font></span></a>, <a class="ref" href="xxreal_1.html#T34" onmouseover="rs('xxreal_1/T34')" onmouseout="rh()">XXREAL_1:34</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="29_4_4_1_2_1_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:29_4_4_1_2_1_4_1_3"/><span class="lab"><font color="Green" title="E66">A82</font></span>: 
<font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">n</font></span>)</span>)</span>)</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">  <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span><br/><div class="add"><a NAME="E2:29_4_4_1_2_1_4_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">Kn</font> <a href="mesfun14.html#K1" title="MESFUN14:func.1">.</a> <font color="Maroon" title="c12">k1</font> <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"><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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font></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="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a></span>;<br/><a NAME="E3:29_4_4_1_2_1_4_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A83</font></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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/><a NAME="E4:29_4_4_1_2_1_4_1_3"/><span class="lab"><font color="Green" title="E68">A84</font></span>: 
<font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E64">A66</font></span></a>, <a class="ref" href="nat_d.html#T36" onmouseover="rs('nat_d/T36')" onmouseout="rh()">NAT_D:36</a>, <a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_4_4_1_2_1_4_1_3_1"><span class="kw">now </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">  not <font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:29_4_4_1_2_1_4_1_3_1"/>
<font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</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"> contradiction</span><br/><a NAME="E2:29_4_4_1_2_1_4_1_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:29_4_4_1_2_1_4_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E68">A84</font></span></a>, <a class="ref" href="nat_d.html#T43" onmouseover="rs('nat_d/T43')" onmouseout="rh()">NAT_D:43</a></span>;<br/><a NAME="E4:29_4_4_1_2_1_4_1_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E5:29_4_4_1_2_1_4_1_3_1"/><span class="kw">then </span>
<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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">k2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <span class="p3">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</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="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:29_4"><span class="lab"><font color="Green" title="E60">A39</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E6:29_4_4_1_2_1_4_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E67">A83</font></span></a>, <a class="txt" href="#E4:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E65">A67</font></span></a>, <a class="txt" href="#E31:29_4"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E66">A82</font></span></a>, <a class="txt" href="#E38:29_4"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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><a NAME="E6:29_4_4_1_2_1_4_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <span class="p2">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:29_4"><span class="lab"><font color="Green" title="E61">A40</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="xreal_1.html#T64" onmouseover="rs('xreal_1/T64')" onmouseout="rh()">XREAL_1:64</a></span>;<br/><a NAME="E7:29_4_4_1_2_1_4_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A85</font></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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:29_4"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="txt" href="#E39:29_4"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E66">A82</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:29_4_4_1_2_1_4_1_3"/><span class="lab"><font color="Green" title="E70">A86</font></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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E5:29_4"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E15:29_4"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E66">A82</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E9:29_4_4_1_2_1_4_1_3"/><span class="lab"><font color="Green" title="E71">A87</font></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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E15:29_4"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="ref" href="integra1.html#D4" onmouseover="rs('integra1/D4')" onmouseout="rh()">INTEGRA1:def 4</a></span>;<br/><a NAME="E10:29_4_4_1_2_1_4_1_3"/><span class="lab"><font color="Green" title="E72">A88</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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>) <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> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <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> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<span class="p4">(<span class="default"><a href="mesfun14.html#K2" title="MESFUN14:func.2">EqDiv</a> (<font color="Maroon" title="c1">A</font>,<span class="p5">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span></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="E11:29_4_4_1_2_1_4_1_3"/><span class="lab"><font color="Green" title="E73">A89</font></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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_4_4_1_2_1_4"><span class="lab"><font color="Green" title="E65">A67</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E12:29_4_4_1_2_1_4_1_3"/>
 <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_4"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E16:29_4"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="integra1.html#T8" onmouseover="rs('integra1/T8')" onmouseout="rh()">INTEGRA1:8</a></span>;<br/><a NAME="E13:29_4_4_1_2_1_4_1_3"/><span class="kw">then </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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font></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="E14:29_4_4_1_2_1_4_1_3"/><span class="kw">then </span>
 <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E72">A88</font></span></a>, <a class="txt" href="#E11:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E73">A89</font></span></a>, <a class="txt" href="#E9:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E71">A87</font></span></a>, <a class="ref" href="xxreal_1.html#T50" onmouseover="rs('xxreal_1/T50')" onmouseout="rh()">XXREAL_1:50</a></span>;<br/><a NAME="E15:29_4_4_1_2_1_4_1_3"/><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="c8">m</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c7">n</font></span>)</span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c12">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E71">A87</font></span></a>, <a class="txt" href="#E1:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E66">A82</font></span></a>, <a class="ref" href="integra1.html#D2" onmouseover="rs('integra1/D2')" onmouseout="rh()">INTEGRA1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E16:29_4_4_1_2_1_4_1_3"/>
 <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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E69">A85</font></span></a>, <a class="txt" href="#E8:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E70">A86</font></span></a>, <a class="txt" href="#E9:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E71">A87</font></span></a>, <a class="txt" href="#E10:29_4_4_1_2_1_4_1_3"><span class="lab"><font color="Green" title="E72">A88</font></span></a>, <a class="ref" href="xxreal_1.html#T34" onmouseover="rs('xxreal_1/T34')" onmouseout="rh()">XXREAL_1:34</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E46:29_4"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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="c8">m</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c13">k2</font>)</span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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="c7">n</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c12">k1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T75" onmouseover="rs('relat_1/T75')" onmouseout="rh()">RELAT_1:75</a></span>;<br/>
<span class="kw">hence </span><a NAME="E47:29_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29_4"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E14:29_4"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="txt" href="#E23:29_4"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="txt" href="#E27:29_4"><span class="lab"><font color="Green" title="E46">A23</font></span></a>, <a class="ref" href="seq_4.html#T48" onmouseover="rs('seq_4/T48')" onmouseout="rh()">SEQ_4:48</a>, <a class="ref" href="relat_1.html#T11" onmouseover="rs('relat_1/T11')" onmouseout="rh()">RELAT_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/>


</div><span class="kw">end;</span></div>
<a NAME="E14:29"/><span class="lab"><font color="Green" title="E33">A90</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>( <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> &amp;  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> ( <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies ( <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> &amp;  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_5"/><span class="lab"><font color="Green" title="E34">A91</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</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"> ( <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> &amp;  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<a NAME="E2:29_5"/>
 for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">m</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">m</font>, <font color="Maroon" title="c9">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="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_5_1"/>
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">m</font></span><br/>

<a NAME="E2:29_5_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_5"><span class="lab"><font color="Green" title="E34">A91</font></span></a>, <a class="txt" href="#E13:29"><span class="lab"><font color="Green" title="E32">A9</font></span></a></span>;<br/>
<a NAME="E3:29_5_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:29_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</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>
<a NAME="E3:29_5"/><span class="kw">then </span>
<font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T7" onmouseover="rs('rinfsup2/T7')" onmouseout="rh()">RINFSUP2:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:29_5"/>
( <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#T36" onmouseover="rs('rinfsup2/T36')" onmouseout="rh()">RINFSUP2:36</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">  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E5:29_5"/>
 for <font color="Olive" title="b1">y</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> holds <br/><font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">y</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> implies <font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_5_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</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"> <font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:29_5_2"/><span class="lab"><font color="Green" title="E35">A92</font></span>: 
( <font color="Maroon" title="c9">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T3" onmouseover="rs('partfun1/T3')" onmouseout="rh()">PARTFUN1:3</a></span>;<br/>
<a NAME="E4:29_5_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_5_2"><span class="lab"><font color="Green" title="E35">A92</font></span></a>, <a class="ref" href="mesfunc5.html#D13" onmouseover="rs('mesfunc5/D13')" onmouseout="rh()">MESFUNC5:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:29_5_2"/>
<font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E1:29_5"><span class="lab"><font color="Green" title="E34">A91</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>
<a NAME="E6:29_5"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D2" onmouseover="rs('xxreal_2/D2')" onmouseout="rh()">XXREAL_2:def 2</a></span>;<br/>
<a NAME="E7:29_5"/><span class="kw">then </span>
 <a href="supinf_2.html#K4" title="SUPINF_2:func.4">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" title="SUPINF_2:func.12">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D4" onmouseover="rs('xxreal_2/D4')" onmouseout="rh()">XXREAL_2:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:29_5"/>
 <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rinfsup2.html#D2" onmouseover="rs('rinfsup2/D2')" onmouseout="rh()">RINFSUP2:def 2</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>
<span class="kw">consider </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E16:29"/><span class="lab"><font color="Green" title="E34">A93</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">and </span><br/><a NAME="E17:29"/><span class="lab"><font color="Green" title="E35">A94</font></span>: 
<font color="Maroon" title="c1">A</font> <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"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></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="measure5.html#T14" onmouseover="rs('measure5/T14')" onmouseout="rh()">MEASURE5:14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">a1</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">b1</font> = <font color="Maroon" title="c8">b</font> as   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D1" onmouseover="rs('xxreal_0/D1')" onmouseout="rh()">XXREAL_0:def 1</a></span>;<br/>
<a NAME="E19:29"/><span class="lab"><font color="Green" title="E36">A95</font></span>: 
 <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b1</font> <a href="xxreal_3.html#K10" title="XXREAL_3:func.10">-</a> <font color="Maroon" title="c9">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:29"><span class="lab"><font color="Green" title="E34">A93</font></span></a>, <a class="txt" href="#E17:29"><span class="lab"><font color="Green" title="E35">A94</font></span></a>, <a class="ref" href="measure5.html#T6" onmouseover="rs('measure5/T6')" onmouseout="rh()">MEASURE5:6</a></span>;<br/>
<a NAME="E20:29"/>
<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K2" title="MEASURE5:func.2">diameter</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur12.html#T71" onmouseover="rs('measur12/T71')" onmouseout="rh()">MEASUR12:71</a></span>;<br/>
<a NAME="E21:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A96</font></span>: 
<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c3">A1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:29"><span class="lab"><font color="Green" title="E36">A95</font></span></a>, <a class="ref" href="xxreal_0.html#T4" onmouseover="rs('xxreal_0/T4')" onmouseout="rh()">XXREAL_0:4</a></span>;<br/>
<a NAME="E22:29"/><span class="lab"><font color="Green" title="E38">A97</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A1</font> holds <br/><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b1">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:29"><span class="lab"><font color="Green" title="E33">A90</font></span></a></span>;<br/>
<a NAME="E23:29"/><span class="lab"><font color="Green" title="E39">A98</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> is <font color="Maroon" title="c3">A1</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="mesfunc2.html#T34" onmouseover="rs('mesfunc2/T34')" onmouseout="rh()">MESFUNC2:34</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">K</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>);<br/>
<a NAME="E24:29"/><span class="lab"><font color="Green" title="E40">A99</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T4" onmouseover="rs('absvalue/T4')" onmouseout="rh()">ABSVALUE:4</a></span>;<br/>
<a NAME="E25:29"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" onmouseover="rs('xxreal_0/T25')" onmouseout="rh()">XXREAL_0:25</a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<a NAME="E26:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A100</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:29"><span class="lab"><font color="Green" title="E40">A99</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E27:29"/><span class="lab"><font color="Green" title="E42">A101</font></span>: 
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#T4" onmouseover="rs('absvalue/T4')" onmouseout="rh()">ABSVALUE:4</a></span>;<br/>
<a NAME="E28:29"/>
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" onmouseover="rs('xxreal_0/T25')" onmouseout="rh()">XXREAL_0:25</a></span>;<br/>
<a NAME="E29:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A102</font></span>: 
 <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:29"><span class="lab"><font color="Green" title="E42">A101</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E30:29"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> holds <br/><span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c13">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> implies <span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:29_6"/><span class="lab"><font color="Green" title="E44">A103</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</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"> <span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">x0</font> = <font color="Maroon" title="c13">x</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E3:29_6"/>
<font color="Maroon" title="c12">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E4:29_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A104</font></span>: 
(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_6"><span class="lab"><font color="Green" title="E44">A103</font></span></a>, <a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E29">A6</font></span></a></span>;<br/>
<a NAME="E5:29_6"/><span class="lab"><font color="Green" title="E46">A105</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="seq_2.html#D8" onmouseover="rs('seq_2/D8')" onmouseout="rh()">SEQ_2:def 8</a>, <a class="ref" href="integra1.html#T11" onmouseover="rs('integra1/T11')" onmouseout="rh()">INTEGRA1:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">K1</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>) as   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a> ;<br/>
<a NAME="E7:29_6"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_6"><span class="lab"><font color="Green" title="E44">A103</font></span></a>, <a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E8:29_6"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_6"><span class="lab"><font color="Green" title="E46">A105</font></span></a>, <a class="ref" href="seq_4.html#D2" onmouseover="rs('seq_4/D2')" onmouseout="rh()">SEQ_4:def 2</a></span>;<br/>
<a NAME="E9:29_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_6"><span class="lab"><font color="Green" title="E45">A104</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:29_6"/><span class="kw">then </span>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p2"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:29"><span class="lab"><font color="Green" title="E41">A100</font></span></a>, <a class="txt" href="#E29:29"><span class="lab"><font color="Green" title="E43">A102</font></span></a>, <a class="txt" href="#E4:29_6"><span class="lab"><font color="Green" title="E45">A104</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:29_6"/><span class="kw">then </span>
(  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <font color="Maroon" title="c15">K1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">K1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D3" onmouseover="rs('xxreal_3/D3')" onmouseout="rh()">XXREAL_3:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:29_6"/>
<span class="p1"><a href="extreal1.html#K1" title="EXTREAL1:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="extreal1.html#K1" title="EXTREAL1:func.1">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>,<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal1.html#T23" onmouseover="rs('extreal1/T23')" onmouseout="rh()">EXTREAL1:23</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>
<a NAME="E31:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A106</font></span>: 
<font color="Maroon" title="c6">F</font> is  <a href="mesfun10.html#V1" title="MESFUN10:attr.1">uniformly_bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun10.html#D1" onmouseover="rs('mesfun10/D1')" onmouseout="rh()">MESFUN10:def 1</a></span>;<br/>
<a NAME="E32:29"/>
 ex <font color="Olive" title="b1">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>) ) &amp; <font color="Olive" title="b1">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c6">F</font></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:29"><span class="lab"><font color="Green" title="E38">A97</font></span></a>, <a class="txt" href="#E23:29"><span class="lab"><font color="Green" title="E39">A98</font></span></a>, <a class="txt" href="#E21:29"><span class="lab"><font color="Green" title="E37">A96</font></span></a>, <a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E31:29"><span class="lab"><font color="Green" title="E44">A106</font></span></a>, <a class="ref" href="mesfun10.html#T19" onmouseover="rs('mesfun10/T19')" onmouseout="rh()">MESFUN10:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:29"/>
 ex <font color="Olive" title="b1">I</font> being   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a> st <br/>( <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="mesfunc2.html#R1" title="MESFUNC2:pred.1">is_simple_func_in</a>  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  &amp;  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K6" title="INTEGRA1:func.6">upper_sum</a> (<font color="Maroon" title="c2">f</font>,<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="b2">n</font></span>)</span>)</span>)</span>) &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>(  <a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> ) ) ) ) &amp; (  for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font> ) &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/>( <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font></span>)</span> &amp;  <a href="rinfsup2.html#K2" title="RINFSUP2:func.2">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#K3" title="MESFUNC5:func.3">#</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font> ) ) &amp;  <a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c6">F</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>  &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">I</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>) ) &amp; <font color="Olive" title="b1">I</font> is  <a href="mesfunc5.html#V10" title="MESFUNC5:attr.10">convergent</a>  &amp;  <a href="mesfunc5.html#K2" title="MESFUNC5:func.2">lim</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><a href="mesfunc8.html#K7" title="MESFUNC8:func.7">lim</a> <font color="Maroon" title="c6">F</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:29"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E12:29"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="#E13:29"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E14:29"><span class="lab"><font color="Green" title="E33">A90</font></span></a>, <a class="txt" href="#E31:29"><span class="lab"><font color="Green" title="E44">A106</font></span></a>, <a class="txt" href="#E22:29"><span class="lab"><font color="Green" title="E38">A97</font></span></a>, <a class="txt" href="#E23:29"><span class="lab"><font color="Green" title="E39">A98</font></span></a>, <a class="txt" href="#E21:29"><span class="lab"><font color="Green" title="E37">A96</font></span></a>, <a class="ref" href="mesfun10.html#T19" onmouseover="rs('mesfun10/T19')" onmouseout="rh()">MESFUN10:19</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>
