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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <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">  <a href="dist_1.html#K10" title="DIST_1:func.10">uniform_distribution</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K8" title="DIST_1:func.8">distribution</a> (<span class="p1">(<span class="default"><a href="dist_1.html#K11" title="DIST_1:func.11">Uniform_FDprobSEQ</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c1">S</font>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">p</font> =  <a href="dist_1.html#K11" title="DIST_1:func.11">Uniform_FDprobSEQ</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">s</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font>;<br/>
<a NAME="E1:47"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font> <a href="dist_1.html#R2" title="DIST_1:pred.2">is_uniformly_distributed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dist_1.html#T26" target="_self" title="DIST_1:th.26">Th26</a></span>;<br/>
<a NAME="E2:47"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
 for <font color="Olive" title="b1">t</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">t</font> <a href="dist_1.html#R2" title="DIST_1:pred.2">is_uniformly_distributed</a>  holds <br/><font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">Finseq-EQclass</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">t</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</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="c4">t</font> <a href="dist_1.html#R2" title="DIST_1:pred.2">is_uniformly_distributed</a>  implies <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">Finseq-EQclass</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1"/>
<font color="Maroon" title="c4">t</font> <a href="dist_1.html#R2" title="DIST_1:pred.2">is_uniformly_distributed</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="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">Finseq-EQclass</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span></span><br/>

<a NAME="E2:47_1"/><span class="kw">then </span>
 <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c4">t</font> <a href="dist_1.html#R1" title="DIST_1:pred.1">-are_prob_equivalent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="dist_1.html#T25" target="_self" title="DIST_1:th.25">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:47_1"/>
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">Finseq-EQclass</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font></span>)</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>
<a NAME="E3:47"/>
( (  for <font color="Olive" title="b1">t</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">Finseq-EQclass</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span> holds <br/><font color="Olive" title="b1">t</font> <a href="dist_1.html#R2" title="DIST_1:pred.2">is_uniformly_distributed</a>  ) &amp;  <a href="dist_1.html#K5" title="DIST_1:func.5">Finseq-EQclass</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dist_1.html#K6" title="DIST_1:func.6">distribution_family</a> <font color="Maroon" title="c1">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dist_1.html#D6" target="_self" title="DIST_1:def.6">Def6</a>, <a class="txt" href="dist_1.html#E39"><span class="lab"><font color="Green" title="E31">Lm2</font></span></a></span>;<br/>
<a NAME="E4:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A3</font></span>: 
 <a href="dist_1.html#K5" title="DIST_1:func.5">Finseq-EQclass</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K10" title="DIST_1:func.10">uniform_distribution</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="dist_1.html#D12" target="_self" title="DIST_1:def.12">Def12</a></span>;<br/>
<a NAME="E5:47"/>
<span class="p1">(<span class="default"><a href="dist_1.html#K7" title="DIST_1:func.7">GenProbSEQ</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="dist_1.html#K5" title="DIST_1:func.5">Finseq-EQclass</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K11" title="DIST_1:func.11">Uniform_FDprobSEQ</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dist_1.html#T12" target="_self" title="DIST_1:th.12">Th12</a></span>;<br/>
<a NAME="E6:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<span class="p1">(<span class="default"><a href="dist_1.html#K7" title="DIST_1:func.7">GenProbSEQ</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dist_1.html#K8" title="DIST_1:func.8">distribution</a> (<span class="p2">(<span class="default"><a href="dist_1.html#K11" title="DIST_1:func.11">Uniform_FDprobSEQ</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c1">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dist_1.html#K7" title="DIST_1:func.7">GenProbSEQ</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="dist_1.html#K5" title="DIST_1:func.5">Finseq-EQclass</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dist_1.html#T22" target="_self" title="DIST_1:th.22">Th22</a></span>;<br/>
<a NAME="E7:47"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="dist_1.html#K7" title="DIST_1:func.7">GenProbSEQ</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K6" title="DIST_1:func.6">distribution_family</a> <font color="Maroon" title="c1">S</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/>
<span class="kw">hence </span><a NAME="E8:47"/>
 <a href="dist_1.html#K10" title="DIST_1:func.10">uniform_distribution</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K8" title="DIST_1:func.8">distribution</a> (<span class="p1">(<span class="default"><a href="dist_1.html#K11" title="DIST_1:func.11">Uniform_FDprobSEQ</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c1">S</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E6:47"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</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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