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<b>set </b><font color="Maroon" title="c39">f</font> =  <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a> ;<br/>
<b>let </b><font color="Maroon" title="c40">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  iff  ex <font color="Olive" title="b1">t</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Olive" title="b1">t</font> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">t</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Olive" title="b1">t</font> implies <font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  )</span>
<div class="add"><b>set </b><font color="Maroon" title="c41">q</font> = <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">p</font>;<br/><b>assume </b><a NAME="E1:45_1"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Olive" title="b1">t</font></span><br/><a NAME="E2:45_1"/><b>then </b><i><font color="Green" title="E44">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c40">p</font>,<span class="p2">(<span class="default"><a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">p</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E3:45_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E4:45_1"/><b>then </b><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c40">p</font> is   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="txt" href="scmbsort.html#E1:45_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="ami_1.html#T125" title="AMI_1:th.125">AMI_1:125</a></i>;<br/><a NAME="E5:45_1"/>
<a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_1.html#K14" title="AMI_1:func.14">FinPartSt</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="txt" href="scmbsort.html#E1:45_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/><a NAME="E6:45_1"/><b>then </b>
<a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">p</font> is   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> 
 <b>by </b><i><a class="ref" href="ami_1.html#T125" title="AMI_1:th.125">AMI_1:125</a></i>;<br/><b>then consider </b><font color="Maroon" title="c42">t</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> , <font color="Maroon" title="c43">u</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> <b> such that </b><br/><a NAME="E8:45_1"/><i><font color="Green" title="E46">A4</font></i>: 
( <font color="Maroon" title="c42">t</font>,<font color="Maroon" title="c43">u</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c43">u</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp; <font color="Maroon" title="c43">u</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> &amp; <font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c42">t</font> &amp; <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c43">u</font> )
 <b>by </b><i><a class="txt" href="scmbsort.html#E2:45_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="scmbsort.html#E4:45_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="scmbsort.html#D3" target="_self" title="SCMBSORT:def.3">Def3</a></i>;<br/><b>take </b><font color="Maroon" title="c44">t</font> = <font color="Maroon" title="c42">t</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c44">t</font></span><br/><b>thus </b><a NAME="E9:45_1"/>
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c44">t</font>
 <b>by </b><i><a class="txt" href="scmbsort.html#E8:45_1"><i><font color="Green" title="E46">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c41">t</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><a NAME="E3:45"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c41">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a> </span><br/>

<b>consider </b><font color="Maroon" title="c42">u</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> <b> such that </b><br/><a NAME="E5:45"/><i><font color="Green" title="E44">A6</font></i>: 
( <font color="Maroon" title="c41">t</font>,<font color="Maroon" title="c42">u</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon" title="c42">u</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp; <font color="Maroon" title="c42">u</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> )
 <b>by </b><i><a class="ref" href="scmbsort.html#T39" target="_self" title="SCMBSORT:th.39">Th39</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c43">u1</font> = <font color="Maroon" title="c42">u</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="txt" href="scmbsort.html#E5:45"><i><font color="Green" title="E44">A6</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c44">q</font> = <span class="p1">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c43">u1</font>;<br/>
<a NAME="E7:45"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c40">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K6" title="SCMFSA_2:func.6">fsloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="scmbsort.html#K1" title="SCMBSORT:func.1">.--&gt;</a> <font color="Maroon" title="c43">u1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a> 
 
<b>by </b><i><a class="txt" href="scmbsort.html#E3:45"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="scmbsort.html#E5:45"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="scmbsort.html#D3" target="_self" title="SCMBSORT:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E8:45"/>
<font color="Maroon" title="c40">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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