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<span class="kw">set </span><font color="Maroon" title="c40">f</font> =  <a href="scmbsort.html#K4" title="SCMBSORT:func.4">Sorting-Function</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c41">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c41">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="c41">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c41">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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="c41">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"><span class="kw">set </span><font color="Maroon" title="c42">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="c41">p</font>;<br/><span class="kw">assume </span><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c41">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c41">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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="E2:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c41">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="c41">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>;<br/><a NAME="E3:39_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="memstr_0.html#K9" title="MEMSTR_0:func.9">FinPartSt</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/><a NAME="E4:39_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A3</font></span>: 
<font color="Maroon" title="c41">p</font> is   <a href="memstr_0.html#NM4" title="MEMSTR_0:NM.4">FinPartState</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="memstr_0.html#T76" title="MEMSTR_0:th.76">MEMSTR_0:76</a></span>;<br/><a NAME="E5:39_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="c41">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="memstr_0.html#K9" title="MEMSTR_0:func.9">FinPartSt</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="ref" href="partfun1.html#T4" title="PARTFUN1:th.4">PARTFUN1:4</a></span>;<br/><a NAME="E6:39_1"/><span class="kw">then </span>
<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="c41">p</font> is   <a href="memstr_0.html#NM4" title="MEMSTR_0:NM.4">FinPartState</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="memstr_0.html#T76" title="MEMSTR_0:th.76">MEMSTR_0:76</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c43">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="c44">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> <span class="kw"> such that </span><br/><a NAME="E8:39_1"/>
<font color="Maroon" title="c43">t</font>,<font color="Maroon" title="c44">u</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">and </span><br/><a NAME="E9:39_1"/>
<font color="Maroon" title="c44">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> 
 <span class="kw">and </span><br/><a NAME="E10:39_1"/>
<font color="Maroon" title="c44">u</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> 
 <span class="kw">and </span><br/><a NAME="E11:39_1"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<font color="Maroon" title="c41">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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">t</font>
 <span class="kw">and </span><br/><a NAME="E12:39_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="c41">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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="c44">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="scmbsort.html#D3" target="_self" title="SCMBSORT:def.3">Def3</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c45">t</font> = <font color="Maroon" title="c43">t</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c41">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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="c45">t</font></span><br/><span class="kw">thus </span><a NAME="E13:39_1"/>
<font color="Maroon" title="c41">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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="c45">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39_1"><span class="lab"><font color="Green" title="E59">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">given </span><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> <span class="kw"> such that </span><a NAME="E3:39"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c41">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c41">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/>

<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E5:39"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
<font color="Maroon" title="c42">t</font>,<font color="Maroon" title="c43">u</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 <span class="kw">and </span><br/><a NAME="E6:39"/><span class="lab"><font color="Green" title="E58">A7</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E7:39"/><span class="lab"><font color="Green" title="E59">A8</font></span>: 
<font color="Maroon" title="c43">u</font> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T33" title="RFINSEQ:th.33">RFINSEQ:33</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c44">u1</font> = <font color="Maroon" title="c43">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E58">A7</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c45">q</font> = <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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">u1</font>;<br/>
<a NAME="E9:39"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c41">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">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">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E59">A8</font></span></a>, <a class="ref" href="scmbsort.html#D3" target="_self" title="SCMBSORT:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:39"/>
<font color="Maroon" title="c41">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 class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T1" title="FUNCT_1:th.1">FUNCT_1:1</a></span>; <a class="txt" 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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