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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>$2 is    <a href="integr16.html#M1" title="INTEGR16:mode.1">middle_volume</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> $1;<br/>
<a NAME="E1:7_1_1"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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#K2" title="NUMBERS:func.2">COMPLEX</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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="c4">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">y</font> =  the    <a href="integr16.html#M1" title="INTEGR16:mode.1">middle_volume</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c4">x</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">y</font> =  the    <a href="integr16.html#M1" title="INTEGR16:mode.1">middle_volume</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c4">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E2:7_1_1_1"/>
<font color="Maroon" title="c6">y</font> is    <a href="integr16.html#M1" title="INTEGR16:mode.1">middle_volume</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Maroon" title="c4">x</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:7_1_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#K2" title="NUMBERS:func.2">COMPLEX</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="c4">x</font>,<font color="Olive" title="b1">y</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:7_1_1"/>
 ex <font color="Olive" title="b1">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> st <br/> for <font color="Olive" title="b2">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f</font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Olive" title="b2">x</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E3:7_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> st <br/> for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K8" title="FINSEQ_2:func.8">.</a> <font color="Olive" title="b2">k</font> is    <a href="integr16.html#M1" title="INTEGR16:mode.1">middle_volume</a> of <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font> <a href="integra2.html#K2" title="INTEGRA2:func.2">.</a> <font color="Olive" title="b2">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"> verum</span><br/>


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