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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/> for <font color="Olive" title="b3">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b4">S</font> being   <a href="integr15.html#M4" title="INTEGR15:mode.4">middle_volume_Sequence</a> of <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">T</font>  st <font color="Olive" title="b2">f</font> is  <a href="integr15.html#V1" title="INTEGR15:attr.1">bounded</a>  &amp; <font color="Olive" title="b2">f</font> is  <a href="integr15.html#V2" title="INTEGR15:attr.2">integrable</a>  &amp;  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b3">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">S</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Olive" title="b2">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/> for <font color="Olive" title="b2">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b3">S</font> being   <a href="integr15.html#M4" title="INTEGR15:mode.4">middle_volume_Sequence</a> of <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">T</font>  st <font color="Olive" title="b1">f</font> is  <a href="integr15.html#V1" title="INTEGR15:attr.1">bounded</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="integr15.html#V2" title="INTEGR15:attr.2">integrable</a>  &amp;  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b2">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">S</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Olive" title="b1">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">S</font> being   <a href="integr15.html#M4" title="INTEGR15:mode.4">middle_volume_Sequence</a> of <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">T</font>  st <font color="Maroon" title="c3">f</font> is  <a href="integr15.html#V1" title="INTEGR15:attr.1">bounded</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="integr15.html#V2" title="INTEGR15:attr.2">integrable</a>  &amp;  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b1">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b2">S</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">T</font> be   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being   <a href="integr15.html#M4" title="INTEGR15:mode.4">middle_volume_Sequence</a> of <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">T</font>  st <font color="Maroon" title="c3">f</font> is  <a href="integr15.html#V1" title="INTEGR15:attr.1">bounded</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="integr15.html#V2" title="INTEGR15:attr.2">integrable</a>  &amp;  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>(  <a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">S</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">S</font> be    <a href="integr15.html#M4" title="INTEGR15:mode.4">middle_volume_Sequence</a> of <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="integr15.html#V1" title="INTEGR15:attr.1">bounded</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="integr15.html#V2" title="INTEGR15:attr.2">integrable</a>  &amp;  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies (  <a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">S</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font> ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
( <font color="Maroon" title="c3">f</font> is  <a href="integr15.html#V1" title="INTEGR15:attr.1">bounded</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="integr15.html#V2" title="INTEGR15:attr.2">integrable</a>  )
 <span class="kw">and </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
(  <a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K3" title="INTEGRA2:func.3">delta</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">S</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">seq</font> =  <a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">xs</font> =  <a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font>;<br/>
<a NAME="E3:32"/>
 <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">xseq</font> =  <a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">S</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E5:32"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b2">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b2">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b2">rseqi</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b2">rseqi</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies  ex <font color="Olive" title="b1">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Olive" title="b1">rseqi</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">rseqi</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">pjinf</font> = <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> $1</span>)</span>;<br/>
<a NAME="E2:32_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E1:32_1_1"/>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c11">x</font></span>)</span> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:32_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c11">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E4:32_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S10" title="FUNCT_2:sch.10">FUNCT_2:sch 10</a>(<a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>);<br/></span>
<a NAME="E5:32_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Olive" title="b1">x</font></span>)</span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/>

<a NAME="E1:32_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:32_1_2"/><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="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 
;<br/>
<span class="kw">then </span> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>
<br/>.= 
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:32_1_2"/>
( <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:32_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> is    <a href="integr15.html#M1" title="INTEGR15:mode.1">middle_volume</a> of <font color="Maroon" title="c10">pjinf</font>,<font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font> is    <a href="integr15.html#M1" title="INTEGR15:mode.1">middle_volume</a> of <font color="Maroon" title="c10">pjinf</font>,<font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">Tk</font> = <font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">Fk</font> = <font color="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E3:32_1_3"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E38">A8</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">Fk</font> = 
 <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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c12">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a></span>
<br/>.= 
 <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="c13">Tk</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D5" target="_self" title="INTEGR15:def.5">Def5</a></span>
;<br/>
<a NAME="E5:32_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">Fk</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="c13">Tk</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E40">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">j</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="c13">Tk</font> implies  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">Fk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> ) )</span><br/><a NAME="E1:32_1_3_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E2:32_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A11</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/><span class="kw">assume </span><a NAME="E3:32_1_3_2"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c15">j</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="c13">Tk</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">Fk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">r</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E5:32_1_3_2"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
<font color="Maroon" title="c16">r</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:32_1_3_2"/><span class="lab"><font color="Green" title="E44">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D5" target="_self" title="INTEGR15:def.5">Def5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">v</font> = <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c18">v</font> = <font color="Maroon" title="c17">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">Fk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c19">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:32_1_3_2"/><span class="lab"><font color="Green" title="E45">A15</font></span>: 
<font color="Maroon" title="c19">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:32_1_3_2"/><span class="lab"><font color="Green" title="E46">A16</font></span>: 
<font color="Maroon" title="c16">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_3_2"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E11:32_1_3_2"/>
<font color="Maroon" title="c19">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_3_2"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E12:32_1_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E13:32_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A17</font></span>: 
<font color="Maroon" title="c19">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_3_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/><span class="lab"><font color="Green" title="E48">A18</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">pjinf</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_3_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
;<br/><span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c16">r</font> = 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_3_2"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="txt" href="#E10:32_1_3_2"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:32_1_3_2"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_3_2"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="txt" href="#E14:32_1_3_2"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
;<br/><span class="kw">hence </span><a NAME="E16:32_1_3_2"/>
<font color="Maroon" title="c18">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">pjinf</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_3_2"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="txt" href="#E14:32_1_3_2"><span class="lab"><font color="Green" title="E48">A18</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">Fk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span></span><br/><span class="kw">thus </span><font color="Maroon" title="c14">Fk</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">j</font> = 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_3"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E5:32_1_3"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E3:32_1_3_2"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c16">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_3_2"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></span>
<br/>.= 
<font color="Maroon" title="c18">v</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<font color="Maroon" title="c15">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:32_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">Fk</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="c13">Tk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_3"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:32_1_3"/>
<font color="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">k</font> is    <a href="integr15.html#M1" title="INTEGR15:mode.1">middle_volume</a> of <font color="Maroon" title="c10">pjinf</font>,<font color="Maroon" title="c4">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_3"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="integr15.html#D1" target="_self" title="INTEGR15:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">pjis</font> = <font color="Maroon" title="c11">F</font> as    <a href="integr15.html#M2" title="INTEGR15:mode.2">middle_volume_Sequence</a> of <font color="Maroon" title="c10">pjinf</font>,<font color="Maroon" title="c4">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D3" target="_self" title="INTEGR15:def.3">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">rseqi</font> =  <a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c10">pjinf</font>,<font color="Maroon" title="c12">pjis</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<span class="kw">assume </span><a NAME="E9:32_1"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">rseqi</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Olive" title="b1">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Olive" title="b1">rseqi</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">rseqi</font> )</span><br/>

<a NAME="E10:32_1"/><span class="lab"><font color="Green" title="E38">A20</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c13">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/>

<a NAME="E1:32_1_4"/><span class="lab"><font color="Green" title="E39">A21</font></span>: 
 ex <font color="Olive" title="b1">Fi</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c14">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integr15.html#K4" title="INTEGR15:func.4">middle_sum</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Olive" title="b1">Fi</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="integr15.html#D6" target="_self" title="INTEGR15:def.6">Def6</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c13">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">k</font> = 
 <a href="integr15.html#K1" title="INTEGR15:func.1">middle_sum</a> <font color="Maroon" title="c10">pjinf</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">pjis</font> <a href="integr15.html#K2" title="INTEGR15:func.2">.</a> <font color="Maroon" title="c14">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D4" target="_self" title="INTEGR15:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integr15.html#K4" title="INTEGR15:func.4">middle_sum</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="integr15.html#K5" title="INTEGR15:func.5">.</a> <font color="Maroon" title="c14">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E1:32_1_4"><span class="lab"><font color="Green" title="E39">A21</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">xseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr15.html#D8" target="_self" title="INTEGR15:def.8">Def8</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">take </span>
<font color="Maroon" title="c13">rseqi</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c13">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c13">rseqi</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c13">rseqi</font> )</span><br/>

<a NAME="E11:32_1"/><span class="lab"><font color="Green" title="E39">A22</font></span>: 
( <span class="p1">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font> is  <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>  &amp; <font color="Maroon" title="c10">pjinf</font> is  <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E9:32_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="integr15.html#D12" target="_self" title="INTEGR15:def.12">Def12</a>, <a class="ref" href="integr15.html#D13" target="_self" title="INTEGR15:def.13">Def13</a></span>;<br/>
<a NAME="E12:32_1"/><span class="kw">then </span>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K3" title="INTEGR15:func.3">middle_sum</a> <font color="Maroon" title="c10">pjinf</font>,<font color="Maroon" title="c12">pjis</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K12" title="INTEGRA1:func.12">integral</a> <font color="Maroon" title="c10">pjinf</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="integr15.html#T9" target="_self" title="INTEGR15:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:32_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c13">rseqi</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">xseq</font> <a href="real_ns1.html#K5" title="REAL_NS1:func.5">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c13">rseqi</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp; <span class="p1">(<span class="default"><a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c13">rseqi</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E9:32_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="txt" href="#E11:32_1"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="txt" href="#E10:32_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="ref" href="integr15.html#D14" target="_self" title="INTEGR15:def.14">Def14</a>, <a class="ref" href="integr15.html#T9" target="_self" title="INTEGR15:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> =  <a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E7:32"/>
 <a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E8:32"/>
(  <a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">S</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><a href="integr15.html#K6" title="INTEGR15:func.6">middle_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integr15.html#K10" title="INTEGR15:func.10">integral</a> <font color="Maroon" title="c3">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="real_ns1.html#T11" title="REAL_NS1:th.11">REAL_NS1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
