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

<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">  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" 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="E11">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_1"/><span class="lab"><font color="Green" title="E12">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" 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="E2:18_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">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" onclick="hs2(this)" href="javascript:()" title="18_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#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#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="18_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:18_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN: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="18_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:18_1_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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">  <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="E2:18_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN: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="uproots.html#K1" title="UPROOTS:func.1">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" title="FUNCT_2:def.3">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">set</a> <span class="kw"> such that </span><br/><a NAME="E4:18_1_1_1_1"/><span class="lab"><font color="Green" title="E13">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="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:18_1_1_1_1"/><span class="lab"><font color="Green" title="E14">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="uproots.html#K1" title="UPROOTS:func.1">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" title="FUNCT_1:def.3">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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_1_1_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/><a NAME="E7:18_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="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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T3" title="UPROOTS:th.3">UPROOTS:3</a></span>;<br/><a NAME="E8:18_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN: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#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_1_1_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E9:18_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c6">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="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" title="DIST_1:def.3">Def3</a></span>;<br/><a NAME="E10:18_1_1_1_1"/>
<font color="Maroon" title="c6">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="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="#E8:18_1_1_1_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="dist_1.html#D3" target="_self" title="DIST_1:def.3">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="#E5:18_1_1_1_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="dist_1.html#D3" target="_self" title="DIST_1:def.3">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="#E1:18_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E5:18_1_1_1_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E9:18_1_1_1_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="dist_1.html#D3" target="_self" title="DIST_1:def.3">Def3</a></span>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:18_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
 not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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">  <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="E2:18_1_1_1_2"/><span class="lab"><font color="Green" title="E14">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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_1_1_1_2_1"/>
<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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:18_1_1_1_2_1"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E4:18_1_1_1_2_1"/>
<font color="Maroon" title="c5">y</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> <font color="Maroon" title="c3">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1_1_1_2_1"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E5:18_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN: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="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<a NAME="E6:18_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 
;<br/>
<a NAME="E7:18_1_1_1_2_1"/>
<font color="Maroon" title="c3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="txt" href="#E3:18_1_1_1_2_1"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:18_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_1_1_2"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="txt" href="#E6:18_1_1_1_2_1"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 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/>


</div><span class="kw">end;</span></div><a NAME="E3:18_1_1_1_2"/>
<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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:18_1_1_1_2_2"/>
<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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:18_1_1_1_2_2"/><span class="lab"><font color="Green" title="E15">A11</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E4:18_1_1_1_2_2"/>
<font color="Maroon" title="c5">y</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> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1_1_1_2_2"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E5:18_1_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN: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="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<a NAME="E6:18_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A12</font></span>: 
<font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 
;<br/>
<a NAME="E7:18_1_1_1_2_2"/>
<font color="Maroon" title="c2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="txt" href="#E3:18_1_1_1_2_2"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:18_1_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_1_1_2"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="txt" href="#E6:18_1_1_1_2_2"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 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/>


</div><span class="kw">end;</span></div><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#K6" title="NUMBERS:func.6">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="#E2:18_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a></span>
;<br/> <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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:18_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dist_1.html#D4" target="_self" title="DIST_1:def.4">Def4</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_2"/><span class="lab"><font color="Green" title="E12">A13</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" 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="E13">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">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="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="uproots.html#K1" title="UPROOTS:func.1">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="E1:18_2_1"/>
<font color="Maroon" title="c4">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="dist_1.html#K4" title="DIST_1:func.4">FDprobSEQ</a> <font color="Maroon" title="c3">t</font></span>)</span>
 ; <a class="txt" 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="uproots.html#K1" title="UPROOTS:func.1">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="uproots.html#K1" title="UPROOTS:func.1">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" title="DIST_1:def.3">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="uproots.html#K1" title="UPROOTS:func.1">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="#E1:18_2"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="ref" href="dist_1.html#D4" target="_self" title="DIST_1:def.4">Def4</a></span>
;<br/> <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="E3:18_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#K5" title="CARD_1:func.5">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" title="DIST_1:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:18_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="#E2:18_2"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="ref" href="dist_1.html#D3" target="_self" title="DIST_1:def.3">Def3</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">hence </span><a NAME="E3:18"/>
( <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="#E1:18"><span class="lab"><font color="Green" title="E11">A1</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>
