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<span class="kw">let </span><font color="Maroon" title="c2">g</font>, <font color="Maroon" title="c3">h</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="dist_1.html#K7" title="DIST_1:func.7">distribution_family</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dist_1.html#K7" title="DIST_1:func.7">distribution_family</a> <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b2">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">FDprobSEQ</a> <font color="Olive" title="b2">s</font> ) ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dist_1.html#K7" title="DIST_1:func.7">distribution_family</a> <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b2">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">FDprobSEQ</a> <font color="Olive" title="b2">s</font> ) ) implies <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_1_2"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dist_1.html#K7" title="DIST_1:func.7">distribution_family</a> <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b2">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">FDprobSEQ</a> <font color="Olive" title="b2">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dist_1.html#K7" title="DIST_1:func.7">distribution_family</a> <font color="Maroon" title="c1">S</font> st <br/>for <font color="Olive" title="b2">s</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/>(  not <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> or  not <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">FDprobSEQ</a> <font color="Olive" title="b2">s</font> ) or <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:20_1_2"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dist_1.html#K7" title="DIST_1:func.7">distribution_family</a> <font color="Maroon" title="c1">S</font>  ex <font color="Olive" title="b2">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">FDprobSEQ</a> <font color="Olive" title="b2">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="dist_1.html#K7" title="DIST_1:func.7">distribution_family</a> <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span> ;<br/><span class="kw">consider </span><font color="Maroon" title="c6">u</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E3:20_1_2_1"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K6" title="DIST_1:func.6">Finseq-EQclass</a> <font color="Maroon" title="c6">u</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></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">t</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E5:20_1_2_1"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E6:20_1_2_1"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">FDprobSEQ</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_2"><span class="lab"><font color="Green" title="E12">A5</font></span></a></span>;<br/><a NAME="E7:20_1_2_1"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c6">u</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="#E3:20_1_2_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E5:20_1_2_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="dist_1.html#T7" target="_self" title="DIST_1:th.7">Th7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E9:20_1_2_1"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E10:20_1_2_1"/><span class="lab"><font color="Green" title="E18">A11</font></span>: 
<font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">FDprobSEQ</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_2"><span class="lab"><font color="Green" title="E11">A4</font></span></a></span>;<br/><a NAME="E11:20_1_2_1"/>
<font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c8">s</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="#E3:20_1_2_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E9:20_1_2_1"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="dist_1.html#T7" target="_self" title="DIST_1:th.7">Th7</a></span>;<br/><a NAME="E12:20_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">t</font>,<font color="Maroon" title="c8">s</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="#E7:20_1_2_1"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="dist_1.html#T6" target="_self" title="DIST_1:th.6">Th6</a></span>;<br/><span class="kw">hence </span><a NAME="E13:20_1_2_1"/>
<font color="Maroon" title="c2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_1_2_1"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="txt" href="#E6:20_1_2_1"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="dist_1.html#T10" target="_self" title="DIST_1:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:20_1_2"/>
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D9" title="FUNCT_2:def.9">FUNCT_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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