<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> ex <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Olive">b<sub>2</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  &amp; <font color="Olive">b<sub>2</sub></font> is <a href="rfinseq.html#V1">non-increasing</a> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<a NAME="E1:54_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>39</sub></font>, <font color="Maroon">c<sub>40</sub></font>, <font color="Maroon">c<sub>41</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>given </b><font color="Maroon">c<sub>42</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> , <font color="Maroon">c<sub>43</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><a NAME="E1:54_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>42</sub></font>,<font color="Maroon">c<sub>43</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon">c<sub>43</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  &amp; <font color="Maroon">c<sub>43</sub></font> is <a href="rfinseq.html#V1">non-increasing</a> &amp; <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">c<sub>42</sub></font> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>43</sub></font> )
 ;<br/>

<b>given </b><font color="Maroon">c<sub>44</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> , <font color="Maroon">c<sub>45</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><a NAME="E2:54_1_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>44</sub></font>,<font color="Maroon">c<sub>45</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon">c<sub>45</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  &amp; <font color="Maroon">c<sub>45</sub></font> is <a href="rfinseq.html#V1">non-increasing</a> &amp; <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">c<sub>44</sub></font> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>45</sub></font> )
 ;<br/>

<font color="Maroon">c<sub>42</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">c<sub>42</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
<font color="Maroon">c<sub>44</sub></font>
<b>by </b><i><a class="txt" href="scmbsort.html#E1:54_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="scmbsort.html#E2:54_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E4:54_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>43</sub></font>,<font color="Maroon">c<sub>45</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="scmbsort.html#E1:54_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="scmbsort.html#E2:54_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/>
<b>hence </b><a NAME="E5:54_1_1_1"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="scmbsort.html#E1:54_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="scmbsort.html#E2:54_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="rfinseq.html#T36">RFINSEQ:36</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>39</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:54_1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>39</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S1">FUNCT_1:sch 1</a>(<a class="txt" href="scmbsort.html#E1:54_1_1"><i><font color="Green">E53</font></i></a>);<br/></i>
<a NAME="E4:54_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>39</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>40</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_1_1_2"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>39</sub></font>
 ;<br/>

<a NAME="E2:54_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>40</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>40</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>hence </b><a NAME="E3:54_1_1_2"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i><a class="txt" href="scmbsort.html#E3:54_1_1"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:54_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>39</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>40</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_1_1_3"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>39</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>41</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54_1_1_3"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>40</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>39</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>42</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> , <font color="Maroon">c<sub>43</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E5:54_1_1_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>42</sub></font>,<font color="Maroon">c<sub>43</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon">c<sub>43</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  &amp; <font color="Maroon">c<sub>43</sub></font> is <a href="rfinseq.html#V1">non-increasing</a> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">c<sub>42</sub></font> &amp; <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>43</sub></font> )
 <b>by </b><i><a class="txt" href="scmbsort.html#E3:54_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="scmbsort.html#E3:54_1_1_3"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>44</sub></font> = <font color="Maroon">c<sub>43</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="txt" href="scmbsort.html#E5:54_1_1_3"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E7:54_1_1_3"/>
<span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">c<sub>44</sub></font> is    <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
;<br/>
<b>hence </b><a NAME="E8:54_1_1_3"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 <b>by </b><i><a class="txt" href="scmbsort.html#E5:54_1_1_3"><i><font color="Green">E57</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>40</sub></font> = <font color="Maroon">c<sub>39</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> , <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a>  <b>by </b><i><a class="txt" href="scmbsort.html#E4:54_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>40</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>41</sub></font>, <font color="Maroon">c<sub>42</sub></font> be    <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>

<b>thus </b><a NAME="E7:54_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>42</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>40</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> ex <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Olive">b<sub>2</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  &amp; <font color="Olive">b<sub>2</sub></font> is <a href="rfinseq.html#V1">non-increasing</a> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="scmbsort.html#E3:54_1_1"><i><font color="Green">E54</font></i></a></i>;<br/>

<b>given </b><font color="Maroon">c<sub>43</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a> , <font color="Maroon">c<sub>44</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><a NAME="E8:54_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>43</sub></font>,<font color="Maroon">c<sub>44</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon">c<sub>44</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K4">INT</a>  &amp; <font color="Maroon">c<sub>44</sub></font> is <a href="rfinseq.html#V1">non-increasing</a> &amp; <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="scmbsort.html#K1" target="_self">.--&gt;</a> <font color="Maroon">c<sub>43</sub></font> &amp; <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6">fsloc</a> 0</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>44</sub></font> )
 ;<br/>

<a NAME="E9:54_1_1"/>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K14">FinPartSt</a> <a href="scmfsa_2.html#K1">SCM+FSA</a> 
 
;<br/>
<b>hence </b><a NAME="E10:54_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>42</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="scmbsort.html#E3:54_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="scmbsort.html#E8:54_1_1"><i><font color="Green">E56</font></i></a></i>;<br/>


</div>
