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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#K7" title="DIST_1:func.7">GenProbSEQ</a> <font color="Maroon" title="c1">S</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</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="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dist_1.html#K6" title="DIST_1:func.6">distribution_family</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dist_1.html#K6" title="DIST_1:func.6">distribution_family</a> <font color="Maroon" title="c1">S</font> &amp; <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> <font color="Maroon" title="c2">x1</font> <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> <font color="Maroon" title="c3">x2</font> implies <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">and </span><br/><a NAME="E2:22_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c3">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">and </span><br/><a NAME="E3:22_1"/><span class="lab"><font color="Green" title="E16">A3</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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x1</font> <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> <font color="Maroon" title="c3">x2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">A1</font> = <font color="Maroon" title="c2">x1</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">u1</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="E6:22_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c4">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">Finseq-EQclass</a> <font color="Maroon" title="c5">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="dist_1.html#D6" target="_self" title="DIST_1:def.6">Def6</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">A2</font> = <font color="Maroon" title="c3">x2</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">u2</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="E9:22_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c6">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dist_1.html#K5" title="DIST_1:func.5">Finseq-EQclass</a> <font color="Maroon" title="c7">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <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="c8">v</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="E11:22_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x2</font>
 <span class="kw">and </span><br/><a NAME="E12:22_1"/><span class="lab"><font color="Green" title="E20">A7</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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x2</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="c8">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="dist_1.html#D7" target="_self" title="DIST_1:def.7">Def7</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">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="E14:22_1"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x1</font>
 <span class="kw">and </span><br/><a NAME="E15:22_1"/><span class="lab"><font color="Green" title="E22">A9</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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x1</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="c9">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="dist_1.html#D7" target="_self" title="DIST_1:def.7">Def7</a></span>;<br/><a NAME="E16:22_1"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c8">v</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="#E3:22_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E15:22_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E12:22_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="dist_1.html#T10" target="_self" title="DIST_1:th.10">Th10</a></span>;<br/><a NAME="E17:22_1"/>
<font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c9">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="#E14:22_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="dist_1.html#T7" target="_self" title="DIST_1:th.7">Th7</a></span>;<br/><a NAME="E18:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c8">v</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="#E16:22_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="dist_1.html#T6" target="_self" title="DIST_1:th.6">Th6</a></span>;<br/><a NAME="E19:22_1"/>
<font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c7">u2</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="#E11:22_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E9:22_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="dist_1.html#T7" target="_self" title="DIST_1:th.7">Th7</a></span>;<br/><a NAME="E20:22_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c7">u2</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="#E18:22_1"><span class="lab"><font color="Green" title="E24">A11</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="E21:22_1"/>
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E9:22_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="dist_1.html#T9" target="_self" title="DIST_1:th.9">Th9</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="E2:22"/>
 <a href="dist_1.html#K7" title="DIST_1:func.7">GenProbSEQ</a> <font color="Maroon" title="c1">S</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</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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