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<span class="kw">let </span><font color="Maroon" title="c1">X1</font>, <font color="Maroon" title="c2">X2</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">  for <font color="Olive" title="b1">S1</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</font><br/>  for <font color="Olive" title="b2">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b3">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>)</span>)</span>  st <font color="Olive" title="b3">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>)</span>)</span> &amp; <font color="Olive" title="b3">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b4">f</font> being   <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font>) ex <font color="Olive" title="b5">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b1">S1</font> ex <font color="Olive" title="b6">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b2">S2</font> ex <font color="Olive" title="b7">Xf</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <br/>( <font color="Olive" title="b3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b4">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">f</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="b4">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">f</font> <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="Olive" title="b5">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">f</font> <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="Olive" title="b6">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">f</font> <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="Olive" title="b7">Xf</font> &amp; (  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b8">n</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="b4">f</font> holds <br/><font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b8">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b6">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b8">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b8">n</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="b7">Xf</font> holds <br/><font color="Olive" title="b7">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b7">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b7">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Olive" title="b3">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b8">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b9">x</font>, <font color="Olive" title="b10">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b8">n</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="b7">Xf</font> &amp; <font color="Olive" title="b9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b10">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b7">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b9">x</font>,<font color="Olive" title="b10">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b5">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b8">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b9">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b6">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b8">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b10">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b8">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Olive" title="b3">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b8">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b7">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b7">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b8">x</font>) ) &amp; (  for <font color="Olive" title="b8">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Olive" title="b3">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b8">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b7">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b7">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b8">y</font>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">S1</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X1</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">S2</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</font><br/>  for <font color="Olive" title="b2">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Olive" title="b1">S2</font>)</span>)</span>  st <font color="Olive" title="b2">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Olive" title="b1">S2</font>)</span>)</span> &amp; <font color="Olive" title="b2">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b3">f</font> being   <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Olive" title="b1">S2</font>) ex <font color="Olive" title="b4">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b5">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b1">S2</font> ex <font color="Olive" title="b6">Xf</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <br/>( <font color="Olive" title="b2">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b3">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">f</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="b3">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">f</font> <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="Olive" title="b4">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">f</font> <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="Olive" title="b5">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">f</font> <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="Olive" title="b6">Xf</font> &amp; (  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b7">n</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="b3">f</font> holds <br/><font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b7">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b7">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b7">n</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="b6">Xf</font> holds <br/><font color="Olive" title="b6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b6">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Olive" title="b2">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b8">x</font>, <font color="Olive" title="b9">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b7">n</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="b6">Xf</font> &amp; <font color="Olive" title="b8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b9">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b8">x</font>,<font color="Olive" title="b9">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b4">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b7">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b8">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b5">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b7">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b9">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b7">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Olive" title="b2">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b7">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b6">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b7">x</font>) ) &amp; (  for <font color="Olive" title="b7">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Olive" title="b2">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b7">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b6">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b7">y</font>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">S2</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X2</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">E</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span>  st <font color="Olive" title="b1">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> &amp; <font color="Olive" title="b1">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b2">f</font> being   <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>) ex <font color="Olive" title="b3">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b4">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b5">Xf</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <br/>( <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b2">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</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="b2">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> <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="Olive" title="b3">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> <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="Olive" title="b4">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">f</font> <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="Olive" title="b5">Xf</font> &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b6">n</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="b2">f</font> holds <br/><font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b6">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b6">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b6">n</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="b5">Xf</font> holds <br/><font color="Olive" title="b5">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b5">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Olive" title="b1">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b7">x</font>, <font color="Olive" title="b8">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b6">n</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="b5">Xf</font> &amp; <font color="Olive" title="b7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b8">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b5">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b7">x</font>,<font color="Olive" title="b8">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b6">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b4">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b6">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b8">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Olive" title="b1">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b6">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b5">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b6">x</font>) ) &amp; (  for <font color="Olive" title="b6">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Olive" title="b1">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b6">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b5">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b6">y</font>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K9" title="PROB_1:func.9">sigma</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</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="c5">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> &amp; <font color="Maroon" title="c5">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  ex <font color="Olive" title="b1">f</font> being   <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>) ex <font color="Olive" title="b2">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b3">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b4">Xf</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <br/>( <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</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">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <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="Olive" title="b2">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <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="Olive" title="b3">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <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="Olive" title="b4">Xf</font> &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</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">f</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b5">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</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="b4">Xf</font> holds <br/><font color="Olive" title="b4">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b4">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b6">x</font>, <font color="Olive" title="b7">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b5">n</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="b4">Xf</font> &amp; <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b6">x</font>,<font color="Olive" title="b7">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b2">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b5">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b4">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b5">x</font>) ) &amp; (  for <font color="Olive" title="b5">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b5">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b4">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b5">y</font>) ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:97"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
( <font color="Maroon" title="c5">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="srings_3.html#K4" title="SRINGS_3:func.4">Field_generated_by</a> <span class="p1">(<span class="default"><a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)</span>)</span> &amp; <font color="Maroon" title="c5">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>) ex <font color="Olive" title="b2">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b3">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b4">Xf</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <br/>( <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</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">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <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="Olive" title="b2">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <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="Olive" title="b3">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <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="Olive" title="b4">Xf</font> &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</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">f</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b5">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b5">n</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="b4">Xf</font> holds <br/><font color="Olive" title="b4">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b4">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b5">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b6">x</font>, <font color="Olive" title="b7">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b5">n</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="b4">Xf</font> &amp; <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b6">x</font>,<font color="Olive" title="b7">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b2">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b5">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b7">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b5">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b4">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b5">x</font>) ) &amp; (  for <font color="Olive" title="b5">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b5">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b4">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b5">y</font>) ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>), <font color="Maroon" title="c7">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c8">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E3:97"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c7">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c8">B</font> &amp; <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">f</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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="c6">f</font> holds <br/>(  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font> ) ) &amp; (  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>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">n</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="c6">f</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="measur11.html#T70" target="_self" onmouseover="rs('measur11/T70')" onmouseout="rh()">Th66</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ) <span class="kw">-&gt; </span>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> =  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">Xf</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E5:97"/><span class="lab"><font color="Green" title="E69">A3</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">Xf</font> <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="c6">f</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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="c9">Xf</font> holds <br/><font color="Maroon" title="c9">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S2" onmouseover="rs('finseq_1/S2')" onmouseout="rh()">FINSEQ_1:sch 2</a>();<br/></span>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_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">  for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">Xf</font> holds <br/><font color="Olive" title="b1">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">z</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="c10">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">Xf</font> implies <font color="Maroon" title="c10">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) )</span><br/><span class="kw">assume </span><a NAME="E1:97_1"/>
<font color="Maroon" title="c10">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">Xf</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="c10">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:97_1"/><span class="lab"><font color="Green" title="E70">A4</font></span>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">Xf</font> &amp; <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c11">i</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a></span>;<br/><a NAME="E5:97_1"/>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E3:97_1"><span class="lab"><font color="Green" title="E70">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:97_1"/>
<font color="Maroon" title="c10">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" onmouseover="rs('funct_2/T8')" onmouseout="rh()">FUNCT_2:8</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>
<a NAME="E7:97"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">Xf</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>)
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">Xf</font> = <font color="Maroon" title="c9">Xf</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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">Xf</font> holds <br/><font color="Maroon" title="c10">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">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="c11">n</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">Xf</font> implies <font color="Maroon" title="c10">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:97_2"/>
<font color="Maroon" title="c11">n</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">Xf</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="c10">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> </span><br/><a NAME="E2:97_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a></span>;<br/><a NAME="E3:97_2"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T39" onmouseover="rs('funct_3/T39')" onmouseout="rh()">FUNCT_3:39</a></span>;<br/><a NAME="E4:97_2"/><span class="kw">then </span>
 not  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E5:97_2"/>
<font color="Maroon" title="c10">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> is  <a href="mesfunc5.html#V3" title="MESFUNC5:attr.3">without-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D3" onmouseover="rs('mesfunc5/D3')" onmouseout="rh()">MESFUNC5:def 3</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="E10:97"/><span class="kw">then </span>
<font color="Maroon" title="c10">Xf</font> is  <a href="measur11.html#V2" title="MEASUR11:attr.2">without_-infty-valued</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">Xf</font> = <font color="Maroon" title="c10">Xf</font> as   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) ;<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">A</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">S1</font> ex <font color="Olive" title="b2">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b3">Xf</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <br/>( <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</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="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="Olive" title="b1">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="Olive" title="b2">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="Olive" title="b3">Xf</font> &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">n</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="c6">f</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b4">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b4">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">n</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="b3">Xf</font> holds <br/><font color="Olive" title="b3">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b3">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b5">x</font>, <font color="Olive" title="b6">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b4">n</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="b3">Xf</font> &amp; <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">x</font>,<font color="Olive" title="b6">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b1">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b4">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b2">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b4">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b6">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b4">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b3">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b4">x</font>) ) &amp; (  for <font color="Olive" title="b4">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b4">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b3">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b4">y</font>) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">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">  ex <font color="Olive" title="b1">B</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">S2</font> ex <font color="Olive" title="b2">Xf</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <br/>( <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</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="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c7">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="Olive" title="b1">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="Olive" title="b2">Xf</font> &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</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="c6">f</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b3">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">n</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="b2">Xf</font> holds <br/><font color="Olive" title="b2">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b2">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b4">x</font>, <font color="Olive" title="b5">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">n</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="b2">Xf</font> &amp; <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b5">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">x</font>,<font color="Olive" title="b5">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Olive" title="b1">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b3">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b5">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b3">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b2">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b3">x</font>) ) &amp; (  for <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b3">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b2">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b3">y</font>) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">B</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">Xf</font> being   <a href="measur11.html#V3" title="MEASUR11:attr.3">summable</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>) st <br/>( <font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</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="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c7">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c8">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="Olive" title="b1">Xf</font> &amp; (  for <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="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="c6">f</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b2">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <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="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">Xf</font> holds <br/><font color="Olive" title="b1">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b1">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">n</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">Xf</font> &amp; <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b4">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b2">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b4">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b2">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b1">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b2">x</font>) ) &amp; (  for <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b2">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Olive" title="b1">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b2">y</font>) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c11">Xf</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="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</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="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c7">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c8">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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">Xf</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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="c6">f</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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">Xf</font> holds <br/><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">n</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">Xf</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>) ) &amp; (  for <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</font>) ) )</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>] <span class="kw">means </span>( $1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> implies <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) );<br/>
<a NAME="E12:97"/><span class="lab"><font color="Green" title="E70">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">Xf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</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="E13:97"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T24" onmouseover="rs('finseq_3/T24')" onmouseout="rh()">FINSEQ_3:24</a></span>;<br/>
<a NAME="E14:97"/><span class="lab"><font color="Green" title="E72">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">k</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">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:97_3"/><span class="lab"><font color="Green" title="E73">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</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">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:97_3"/><span class="lab"><font color="Green" title="E74">A8</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <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"> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_3_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="E1:97_3_1"/>
( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c12">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:97_3_1_1"/><span class="lab"><font color="Green" title="E75">A10</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A11</font></span>: <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>
<br/>.= 
 <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E2:97_3"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E12:97"><span class="lab"><font color="Green" title="E70">A9</font></span></a></span>
;<br/><a NAME="E3:97_3_1_1"/>
 not <font color="Maroon" title="c6">f</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_3"><span class="lab"><font color="Green" title="E74">A8</font></span></a></span>;<br/><a NAME="E4:97_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_3_1_1"><span class="lab"><font color="Green" title="E75">A10</font></span></a>, <a class="ref" href="finseq_5.html#T20" onmouseover="rs('finseq_5/T20')" onmouseout="rh()">FINSEQ_5:20</a></span>;<br/><a NAME="E5:97_3_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" onmouseover="rs('finseq_1/T39')" onmouseout="rh()">FINSEQ_1:39</a></span>;<br/><a NAME="E6:97_3_1_1"/><span class="kw">then </span>
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T25" onmouseover="rs('zfmisc_1/T25')" onmouseout="rh()">ZFMISC_1:25</a></span>;<br/><span class="kw">hence </span><a NAME="E7:97_3_1_1"/>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_3_1_1"><span class="lab"><font color="Green" title="E76">A11</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:97_3_1_2"/>
<font color="Maroon" title="c12">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span><br/><div class="add"><a NAME="E2:97_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A12</font></span>: 
<font color="Maroon" title="c12">k</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="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/><a NAME="E3:97_3_1_2"/><span class="lab"><font color="Green" title="E76">A13</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">k</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_3"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</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="E4:97_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A14</font></span>: 
<font color="Maroon" title="c12">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> <font color="Maroon" title="c11">Xf</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="E5:97_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/><a NAME="E6:97_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">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> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_3_1_2"><span class="lab"><font color="Green" title="E75">A12</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="E7:97_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A16</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <font color="Maroon" title="c12">k</font> <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E4:97_3_1_2"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="txt" href="#E1:97_3"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="txt" href="#E2:97_3_1_2"><span class="lab"><font color="Green" title="E75">A12</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="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><span class="lab"><font color="Green" title="E79">A17</font></span>: <font color="Maroon" title="c11">Xf</font> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_3"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E12:97"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_3"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="txt" href="#E12:97"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a></span>
;<br/><div><span class="lab"><font color="Green" title="E80">A24</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_3_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 <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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></a><div class="add"><span class="kw">assume </span><a NAME="E1:97_3_1_2_2"/>
<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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> 
 ; <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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:97_3_1_2_2"/><span class="lab"><font color="Green" title="E81">A18</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><a NAME="E4:97_3_1_2_2"/><span class="lab"><font color="Green" title="E82">A19</font></span>: 
( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97_3_1_2_2"><span class="lab"><font color="Green" title="E81">A18</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:97_3_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E7:97_3_1_2_2"/><span class="lab"><font color="Green" title="E83">A20</font></span>: 
( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Z</font> &amp; <font color="Maroon" title="c14">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">j</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="E9:97_3_1_2_2"/><span class="lab"><font color="Green" title="E84">A21</font></span>: 
( <font color="Maroon" title="c15">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span> &amp; <font color="Maroon" title="c14">Z</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="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:97_3_1_2_2"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="ref" href="partfun1.html#T3" onmouseover="rs('partfun1/T3')" onmouseout="rh()">PARTFUN1:3</a></span>;<br/><a NAME="E10:97_3_1_2_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:97_3_1_2_2"><span class="lab"><font color="Green" title="E84">A21</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="E11:97_3_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A22</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_3_1_2"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><a NAME="E12:97_3_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A23</font></span>: 
<font color="Maroon" title="c14">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:97_3_1_2_2"><span class="lab"><font color="Green" title="E84">A21</font></span></a>, <a class="ref" href="finseq_3.html#T112" onmouseover="rs('finseq_3/T112')" onmouseout="rh()">FINSEQ_3:112</a></span>;<br/><a NAME="E13:97_3_1_2_2"/>
<font color="Maroon" title="c15">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:97_3_1_2_2"><span class="lab"><font color="Green" title="E85">A22</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="E14:97_3_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_2.html#D2" onmouseover="rs('prob_2/D2')" onmouseout="rh()">PROB_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E15:97_3_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:97_3_1_2_2"><span class="lab"><font color="Green" title="E86">A23</font></span></a>, <a class="txt" href="#E4:97_3_1_2_2"><span class="lab"><font color="Green" title="E82">A19</font></span></a>, <a class="txt" href="#E7:97_3_1_2_2"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:97_3_1_2"/><span class="lab"><font color="Green" title="E81">A25</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="measur11.html#K14" title="MEASUR11:func.14">+</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:97_3_1_2"><span class="lab"><font color="Green" title="E78">A16</font></span></a>, <a class="txt" href="#E8:97_3_1_2"><span class="lab"><font color="Green" title="E79">A17</font></span></a>, <a class="txt" href="#E2:97_3_1_2"><span class="lab"><font color="Green" title="E75">A12</font></span></a>, <a class="txt" href="#E3:97_3_1_2"><span class="lab"><font color="Green" title="E76">A13</font></span></a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E11:97_3_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">k</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97_3_1_2"><span class="lab"><font color="Green" title="E76">A13</font></span></a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/><a NAME="E12:97_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E13:97_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A26</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_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">  for <font color="Olive" title="b1">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">z</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E1:97_3_1_2_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>,<span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="measur11.html#K14" title="MEASUR11:func.14">+</a> <span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:97_3_1_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E13:97_3_1_2"><span class="lab"><font color="Green" title="E82">A26</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E2:97_3_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A28</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>,<span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">z</font></span>)</span> <a href="xxreal_3.html#K9" title="XXREAL_3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:97_3_1_2"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="ref" href="mesfunc1.html#D3" onmouseover="rs('mesfunc1/D3')" onmouseout="rh()">MESFUNC1:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_3_1_2_3_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="E1:97_3_1_2_3_1"/>
( <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> or  not <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_3_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:97_3_1_2_3_1_1"/><span class="lab"><font color="Green" title="E84">A31</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:97_3_1_2_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E4:97_3_1_2_3_1_1"/><span class="lab"><font color="Green" title="E85">A29</font></span>: 
( <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">Z</font> &amp; <font color="Maroon" title="c14">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">j</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="E6:97_3_1_2_3_1_1"/><span class="lab"><font color="Green" title="E86">A30</font></span>: 
( <font color="Maroon" title="c15">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">Z</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="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_3_1_2_3_1_1"><span class="lab"><font color="Green" title="E85">A29</font></span></a>, <a class="ref" href="partfun1.html#T3" onmouseover="rs('partfun1/T3')" onmouseout="rh()">PARTFUN1:3</a></span>;<br/><a NAME="E7:97_3_1_2_3_1_1"/><span class="lab"><font color="Green" title="E87">A36</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:97_3_1_2_3_1_1"><span class="lab"><font color="Green" title="E86">A30</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="E8:97_3_1_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A32</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97_3_1_2"><span class="lab"><font color="Green" title="E76">A13</font></span></a>, <a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><a NAME="E9:97_3_1_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A33</font></span>: 
<font color="Maroon" title="c14">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:97_3_1_2_3_1_1"><span class="lab"><font color="Green" title="E86">A30</font></span></a>, <a class="ref" href="finseq_3.html#T112" onmouseover="rs('finseq_3/T112')" onmouseout="rh()">FINSEQ_3:112</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_3_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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_3_1_2_3_1_1_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="E1:97_3_1_2_3_1_1_1_1"/>
( <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c15">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_3_1_2_3_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:97_3_1_2_3_1_1_1_1_1"/>
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:97_3_1_2_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A34</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_3_1_2_3_1_1"><span class="lab"><font color="Green" title="E85">A29</font></span></a>, <a class="txt" href="#E6:97_3_1_2_3_1_1"><span class="lab"><font color="Green" title="E86">A30</font></span></a>, <a class="ref" href="finseq_3.html#T112" onmouseover="rs('finseq_3/T112')" onmouseout="rh()">FINSEQ_3:112</a></span>;<br/><a NAME="E3:97_3_1_2_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A35</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/><a NAME="E4:97_3_1_2_3_1_1_1_1_1"/>
 not <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:97_3_1_2"><span class="lab"><font color="Green" title="E80">A24</font></span></a>, <a class="txt" href="#E2:97_3_1_2_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E90">A34</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:97_3_1_2_3_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:97_3_1_2_3_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_3_1_2_3"><span class="lab"><font color="Green" title="E83">A28</font></span></a>, <a class="txt" href="#E3:97_3_1_2_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E91">A35</font></span></a>, <a class="ref" href="xxreal_3.html#T4" onmouseover="rs('xxreal_3/T4')" onmouseout="rh()">XXREAL_3:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_3_1_2_3_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:97_3_1_2_3_1_1_1_1_2"/>
<font color="Maroon" title="c15">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:97_3_1_2_3_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:97_3_1_2_3_1_1"><span class="lab"><font color="Green" title="E88">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="E3:97_3_1_2_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A37</font></span>: 
<font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</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="E4:97_3_1_2_3_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E4:97_3_1_2"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><a NAME="E5:97_3_1_2_3_1_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c15">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span> &amp; <font color="Maroon" title="c14">Z</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="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:97_3_1_2_3_1_1"><span class="lab"><font color="Green" title="E89">A33</font></span></a>, <a class="txt" href="#E7:97_3_1_2_3_1_1"><span class="lab"><font color="Green" title="E87">A36</font></span></a>, <a class="txt" href="#E3:97_3_1_2_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E90">A37</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="finseq_3.html#T112" onmouseover="rs('finseq_3/T112')" onmouseout="rh()">FINSEQ_3:112</a></span>;<br/><a NAME="E6:97_3_1_2_3_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E7:97_3_1_2_3_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_3_1_2_3_1_1"><span class="lab"><font color="Green" title="E85">A29</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><a NAME="E8:97_3_1_2_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A38</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><a NAME="E9:97_3_1_2_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A39</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/><a NAME="E10:97_3_1_2_3_1_1_1_1_2"/>
 not <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:97_3_1_2"><span class="lab"><font color="Green" title="E80">A24</font></span></a>, <a class="txt" href="#E8:97_3_1_2_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E91">A38</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E11:97_3_1_2_3_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E12:97_3_1_2_3_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_3_1_2_3"><span class="lab"><font color="Green" title="E83">A28</font></span></a>, <a class="txt" href="#E9:97_3_1_2_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E92">A39</font></span></a>, <a class="ref" href="xxreal_3.html#T4" onmouseover="rs('xxreal_3/T4')" onmouseout="rh()">XXREAL_3:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:97_3_1_2_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_3_1_2_3_1_1"><span class="lab"><font color="Green" title="E84">A31</font></span></a>, <a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</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="97_3_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:97_3_1_2_3_1_2"/><span class="lab"><font color="Green" title="E84">A40</font></span>: 
 not <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:97_3_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A41</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/><a NAME="E3:97_3_1_2_3_1_2"/><span class="lab"><font color="Green" title="E86">A42</font></span>: 
 not <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_3_1_2_3_1_2"><span class="lab"><font color="Green" title="E84">A40</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><a NAME="E4:97_3_1_2_3_1_2"/><span class="lab"><font color="Green" title="E87">A43</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 holds <br/> not <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">j</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies  not <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:97_3_1_2_3_1_2_1"/><span class="lab"><font color="Green" title="E88">B1</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.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">  not <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font></span><br/>

<a NAME="E2:97_3_1_2_3_1_2_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Maroon" title="c14">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E3:97_3_1_2"><span class="lab"><font color="Green" title="E76">A13</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/>
<a NAME="E3:97_3_1_2_3_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:97_3_1_2_3_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E5:97_3_1_2_3_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_3_1_2_3_1_2_1"><span class="lab"><font color="Green" title="E88">B1</font></span></a>, <a class="ref" href="finseq_3.html#T112" onmouseover="rs('finseq_3/T112')" onmouseout="rh()">FINSEQ_3:112</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:97_3_1_2_3_1_2_1"/>
 not <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97_3_1_2_3_1_2"><span class="lab"><font color="Green" title="E86">A42</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_3_1_2_3_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">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:97_3_1_2_3_1_2_2"/>
<font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</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"> contradiction</span><br/><a NAME="E2:97_3_1_2_3_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E4:97_3_1_2_3_1_2_2"/><span class="lab"><font color="Green" title="E88">A46</font></span>: 
( <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">Z</font> &amp; <font color="Maroon" title="c14">Z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">j</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="E6:97_3_1_2_3_1_2_2"/><span class="lab"><font color="Green" title="E89">A47</font></span>: 
( <font color="Maroon" title="c15">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span> &amp; <font color="Maroon" title="c14">Z</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="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_3_1_2_3_1_2_2"><span class="lab"><font color="Green" title="E88">A46</font></span></a>, <a class="ref" href="partfun1.html#T3" onmouseover="rs('partfun1/T3')" onmouseout="rh()">PARTFUN1:3</a></span>;<br/><a NAME="E7:97_3_1_2_3_1_2_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:97_3_1_2_3_1_2_2"><span class="lab"><font color="Green" title="E89">A47</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="E8:97_3_1_2_3_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A48</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E4:97_3_1_2"><span class="lab"><font color="Green" title="E77">A14</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><a NAME="E9:97_3_1_2_3_1_2_2"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E10:97_3_1_2_3_1_2_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_3_1_2_3_1_2"><span class="lab"><font color="Green" title="E87">A43</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:97_3_1_2_3_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_3_1_2_3_1_2_2"><span class="lab"><font color="Green" title="E88">A46</font></span></a>, <a class="txt" href="#E6:97_3_1_2_3_1_2_2"><span class="lab"><font color="Green" title="E89">A47</font></span></a>, <a class="txt" href="#E8:97_3_1_2_3_1_2_2"><span class="lab"><font color="Green" title="E90">A48</font></span></a>, <a class="ref" href="finseq_3.html#T112" onmouseover="rs('finseq_3/T112')" onmouseout="rh()">FINSEQ_3:112</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="E6:97_3_1_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A50</font></span>: 
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/><a NAME="E7:97_3_1_2_3_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E8:97_3_1_2_3_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c13">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:97_3_1_2_3_1_2"><span class="lab"><font color="Green" title="E87">A43</font></span></a></span>;<br/><a NAME="E9:97_3_1_2_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D3" onmouseover="rs('funct_3/D3')" onmouseout="rh()">FUNCT_3:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E10:97_3_1_2_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:97_3_1_2_3_1_2"><span class="lab"><font color="Green" title="E85">A41</font></span></a>, <a class="txt" href="#E2:97_3_1_2_3"><span class="lab"><font color="Green" title="E83">A28</font></span></a>, <a class="txt" href="#E6:97_3_1_2_3_1_2"><span class="lab"><font color="Green" title="E88">A50</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E15:97_3_1_2"/>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:97_3_1_2"><span class="lab"><font color="Green" title="E82">A26</font></span></a>, <a class="ref" href="funct_2.html#D8" onmouseover="rs('funct_2/D8')" onmouseout="rh()">FUNCT_2:def 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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E15:97"/><span class="lab"><font color="Green" title="E73">A51</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">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E13:97"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="txt" href="#E14:97"><span class="lab"><font color="Green" title="E72">A6</font></span></a>);<br/></span>
<span class="kw">thus </span><a NAME="E16:97"/>
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E68">A2</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</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="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c7">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c8">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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">Xf</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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="c6">f</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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">Xf</font> holds <br/><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">n</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">Xf</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>) ) &amp; (  for <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</font>) ) )</span><br/>

<a NAME="E17:97"/>
 <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="c6">f</font></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="txt" href="#E1:97"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<a NAME="E18:97"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <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="zfmisc_1.html#T2" onmouseover="rs('zfmisc_1/T2')" onmouseout="rh()">ZFMISC_1:2</a>, <a class="ref" href="relat_1.html#T42" onmouseover="rs('relat_1/T42')" onmouseout="rh()">RELAT_1:42</a></span>;<br/>
<a NAME="E19:97"/><span class="kw">then </span>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></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="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E20:97"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A52</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T7" onmouseover="rs('finseq_3/T7')" onmouseout="rh()">FINSEQ_3:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:97"/><span class="lab"><font color="Green" title="E75">A53</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</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="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c7">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c8">B</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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">Xf</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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="c6">f</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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">Xf</font> holds <br/><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">n</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">Xf</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>) ) &amp; (  for <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</font>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E22:97"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c7">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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="c8">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E68">A2</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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">Xf</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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="c6">f</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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">Xf</font> holds <br/><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">n</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">Xf</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>) ) &amp; (  for <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</font>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E23:97"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <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">Xf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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="c6">f</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> ) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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">Xf</font> holds <br/><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">n</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">Xf</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>) ) &amp; (  for <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</font>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E24:97"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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="c6">f</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.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"> ( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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">Xf</font> holds <br/><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) ) &amp; <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">n</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">Xf</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>) ) &amp; (  for <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</font>) ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_4"><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"> ( <font color="Maroon" title="c12">n</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="c6">f</font> implies <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:97_4"/><span class="lab"><font color="Green" title="E76">A54</font></span>: 
<font color="Maroon" title="c12">n</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="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"> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span><br/>

<a NAME="E2:97_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="measur10.html#K3" title="MEASUR10:func.3">measurable_rectangles</a> (<font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T4" onmouseover="rs('partfun1/T4')" onmouseout="rh()">PARTFUN1:4</a></span>;<br/>
<a NAME="E3:97_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> where <font color="Olive" title="b1">A</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Olive" title="b2">B</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur10.html#D5" onmouseover="rs('measur10/D5')" onmouseout="rh()">MEASUR10:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">An</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c14">Bn</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E5:97_4"/><span class="lab"><font color="Green" title="E77">A55</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c13">An</font>,<font color="Maroon" title="c14">Bn</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_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="E1:97_4_1"/>
( <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:97_4_1_1"/><span class="lab"><font color="Green" title="E78">A57</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span><br/><div class="add"><a NAME="E2:97_4_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> &amp; <font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E1:97_4"><span class="lab"><font color="Green" title="E76">A54</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:97_4_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_4_1_1"><span class="lab"><font color="Green" title="E78">A57</font></span></a>, <a class="ref" href="zfmisc_1.html#T90" onmouseover="rs('zfmisc_1/T90')" onmouseout="rh()">ZFMISC_1:90</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="97_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:97_4_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span><br/><div class="add"><a NAME="E2:97_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A59</font></span>: 
(  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">An</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Bn</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97_4"><span class="lab"><font color="Green" title="E77">A55</font></span></a>, <a class="ref" href="funct_5.html#T9" onmouseover="rs('funct_5/T9')" onmouseout="rh()">FUNCT_5:9</a></span>;<br/><a NAME="E3:97_4_1_2"/>
(  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E1:97_4"><span class="lab"><font color="Green" title="E76">A54</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:97_4_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97_4"><span class="lab"><font color="Green" title="E77">A55</font></span></a>, <a class="txt" href="#E2:97_4_1_2"><span class="lab"><font color="Green" title="E78">A59</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E25:97"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</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">Xf</font> holds <br/><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>) &amp; (  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>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">n</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">Xf</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>) ) &amp; (  for <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</font>) ) )</span><br/>

<span class="lab"><font color="Green" title="E76">A60</font></span>: <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span> = 
 <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:97"><span class="lab"><font color="Green" title="E73">A51</font></span></a>, <a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="txt" href="#E21:97"><span class="lab"><font color="Green" title="E75">A53</font></span></a></span>
<br/>.= 
 <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T58" onmouseover="rs('finseq_1/T58')" onmouseout="rh()">FINSEQ_1:58</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E27:97"/>
<span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E68">A2</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"> ( (  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>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">n</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">Xf</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>) ) &amp; (  for <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</font>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E28:97"/>
 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>, <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">n</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">Xf</font> &amp; <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b3">y</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"> ( (  for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font>) ) &amp; (  for <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</font>) ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_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>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Maroon" title="c12">n</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">Xf</font> &amp; <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</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="c12">n</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">Xf</font> &amp; <font color="Maroon" title="c13">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Maroon" title="c14">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c14">y</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:97_6"/><span class="lab"><font color="Green" title="E77">Q1</font></span>: 
( <font color="Maroon" title="c12">n</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">Xf</font> &amp; <font color="Maroon" title="c13">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X1</font> &amp; <font color="Maroon" title="c14">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">X2</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="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c14">y</font></span>)</span></span><br/>

<a NAME="E2:97_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c14">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="txt" href="#E12:97"><span class="lab"><font color="Green" title="E70">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:97_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</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="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c1">X1</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="xxreal_3.html#K11" title="XXREAL_3:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="prob_3.html#K8" title="PROB_3:func.8">.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c2">X2</font>)</span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c14">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:97_6"><span class="lab"><font color="Green" title="E77">Q1</font></span></a>, <a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E29:97"/>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font> holds   <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</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">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font> holds   <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Olive" title="b1">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Olive" title="b1">y</font>)</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="97_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Maroon" title="c12">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">x</font>)</span><br/>

<a NAME="E1:97_7"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</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="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/>
<a NAME="E2:97_7"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</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> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:97"><span class="lab"><font color="Green" title="E74">A52</font></span></a>, <a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:97_7"/>
 <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Maroon" title="c12">x</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K5" title="MESFUNC9:func.5">ProjMap1</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:97"><span class="lab"><font color="Green" title="E76">A60</font></span></a>, <a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</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">let </span><font color="Maroon" title="c12">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Maroon" title="c12">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">y</font>)</span><br/>

<a NAME="E30:97"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</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="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="ref" href="measur11.html#D11" target="_self" onmouseover="rs('measur11/D11')" onmouseout="rh()">DEF13</a></span>;<br/>
<a NAME="E31:97"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</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> <span class="p1">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:97"><span class="lab"><font color="Green" title="E74">A52</font></span></a>, <a class="txt" href="#E5:97"><span class="lab"><font color="Green" title="E69">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:97"/>
 <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><a href="measur11.html#K1" title="MEASUR11:func.1">chi</a> (<font color="Maroon" title="c5">E</font>,<span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X1</font>,<font color="Maroon" title="c2">X2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>)</span>)</span>,<font color="Maroon" title="c12">y</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="mesfunc9.html#K6" title="MESFUNC9:func.6">ProjMap2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measur11.html#K12" title="MEASUR11:func.12">Partial_Sums</a> <font color="Maroon" title="c11">Xf</font></span>)</span> <a href="measur11.html#K11" title="MEASUR11:func.11">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Xf</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:97"><span class="lab"><font color="Green" title="E76">A60</font></span></a>, <a class="txt" href="#E3:97"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</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>
