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

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font> <a href="dist_1.html#R1" title="DIST_1:pred.1">-are_prob_equivalent</a>  iff  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Olive" title="b2">t</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">s</font>, <font color="Maroon" title="c3">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" 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"> ( <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font> <a href="dist_1.html#R1" title="DIST_1:pred.1">-are_prob_equivalent</a>  iff  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font> )</span><br/>



<div><span class="lab"><font color="Green" title="E8">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span> <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"> (  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font> implies <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font> <a href="dist_1.html#R1" title="DIST_1:pred.1">-are_prob_equivalent</a>  )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
 <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</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"> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font> <a href="dist_1.html#R1" title="DIST_1:pred.1">-are_prob_equivalent</a> </span><br/><a NAME="E2:19_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">s</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_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="hidden.html#M2" title="HIDDEN:mode.2">set</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">  <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">s</font>)</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:19_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">S</font> or  not <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:19_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">S</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">  <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">s</font>)</span><br/><div class="add"><a NAME="E2:19_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:19_1_1_1_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:19_1_1_1_1"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c5">n</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a></span>;<br/><a NAME="E7:19_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T93" onmouseover="rs('finseq_1/T93')" onmouseout="rh()">FINSEQ_1:93</a></span>;<br/><a NAME="E8:19_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E9:19_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dist_1.html#D3" target="_self" onmouseover="rs('dist_1/D3')" onmouseout="rh()">Def3</a></span>;<br/><a NAME="E10:19_1_1_1_1"/>
<font color="Maroon" title="c6">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c2">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:19_1_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="dist_1.html#D3" target="_self" onmouseover="rs('dist_1/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">hence </span> <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">s</font>) = 
<span class="p1">(<span class="default"><a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1_1_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="dist_1.html#D3" target="_self" onmouseover="rs('dist_1/D3')" onmouseout="rh()">Def3</a></span>
<br/>.= 
 <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">t</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E5:19_1_1_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E9:19_1_1_1_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="dist_1.html#D3" target="_self" onmouseover="rs('dist_1/D3')" onmouseout="rh()">Def3</a></span>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:19_1_1_1_2"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
 not <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">S</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">  <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">s</font>)</span><br/><div class="add"><a NAME="E2:19_1_1_1_2"/>
 not <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a></span>;<br/><a NAME="E3:19_1_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">t</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T50" onmouseover="rs('zfmisc_1/T50')" onmouseout="rh()">ZFMISC_1:50</a></span>;<br/><a NAME="E4:19_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T138" onmouseover="rs('relat_1/T138')" onmouseout="rh()">RELAT_1:138</a></span>;<br/><a NAME="E5:19_1_1_1_2"/>
 not <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1_1_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a></span>;<br/><a NAME="E6:19_1_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">s</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T50" onmouseover="rs('zfmisc_1/T50')" onmouseout="rh()">ZFMISC_1:50</a></span>;<br/><a NAME="E7:19_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">s</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T138" onmouseover="rs('relat_1/T138')" onmouseout="rh()">RELAT_1:138</a></span>;<br/><span class="kw">hence </span> <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">s</font>) = 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 

<br/>.= 
 <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">t</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1_1_2"><span class="lab"><font color="Green" title="E11">A8</font></span></a></span>
;<br/> <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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:19_1"/>
<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font> <a href="dist_1.html#R1" title="DIST_1:pred.1">-are_prob_equivalent</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"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">now </span> <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"> ( <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font> <a href="dist_1.html#R1" title="DIST_1:pred.1">-are_prob_equivalent</a>  implies  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_2"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
<font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font> <a href="dist_1.html#R1" title="DIST_1:pred.1">-are_prob_equivalent</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">  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font></span><br/><div><span class="lab"><font color="Green" title="E10">A10</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_2_1"><span class="kw">now </span> <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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c2">s</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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"> ( <font color="Maroon" title="c4">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font></span>)</span> implies <span class="p1">(<span class="default"><a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>,<font color="Maroon" title="c2">s</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:19_2_1"/>
<font color="Maroon" title="c4">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font></span>)</span>
 ; <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"> <span class="p1">(<span class="default"><a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>,<font color="Maroon" title="c2">s</font>)</span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> = 
 <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>,<font color="Maroon" title="c3">t</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="dist_1.html#D3" target="_self" onmouseover="rs('dist_1/D3')" onmouseout="rh()">Def3</a></span>
<br/>.= 
 <a href="dist_1.html#K3" title="DIST_1:func.3">FDprobability</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>,<font color="Maroon" title="c2">s</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_2"><span class="lab"><font color="Green" title="E9">A9</font></span></a></span>
;<br/> <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="E3:19_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dist_1.html#D3" target="_self" onmouseover="rs('dist_1/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:19_2"/>
 <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_2"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="ref" href="dist_1.html#D3" target="_self" onmouseover="rs('dist_1/D3')" onmouseout="rh()">Def3</a></span>; <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>
<span class="kw">hence </span><a NAME="E3:19"/>
( <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c3">t</font> <a href="dist_1.html#R1" title="DIST_1:pred.1">-are_prob_equivalent</a>  iff  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>; <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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