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<span class="kw">let </span><font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c2">Omega</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">Sigma</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">Omega</font><br/>  for <font color="Olive" title="b2">P</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Olive" title="b1">Sigma</font><br/>  for <font color="Olive" title="b3">F</font> being    <a href="kolmog01.html#M1" title="KOLMOG01:mode.1">ManySortedSigmaField</a> of <font color="Maroon" title="c1">I</font>,<font color="Olive" title="b1">Sigma</font><br/>  for <font color="Olive" title="b4">J</font>, <font color="Olive" title="b5">K</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b3">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b4">J</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Olive" title="b5">K</font> holds <br/> for <font color="Olive" title="b6">a</font>, <font color="Olive" title="b7">c</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font>  st <font color="Olive" title="b6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Olive" title="b4">J</font>,<font color="Olive" title="b3">F</font>) &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Olive" title="b5">K</font>,<font color="Olive" title="b3">F</font>) holds <br/><font color="Olive" title="b2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">a</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">c</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Sigma</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">Omega</font>; <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">P</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c3">Sigma</font><br/>  for <font color="Olive" title="b2">F</font> being    <a href="kolmog01.html#M1" title="KOLMOG01:mode.1">ManySortedSigmaField</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">Sigma</font><br/>  for <font color="Olive" title="b3">J</font>, <font color="Olive" title="b4">K</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b2">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b3">J</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Olive" title="b4">K</font> holds <br/> for <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">c</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Olive" title="b3">J</font>,<font color="Olive" title="b2">F</font>) &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Olive" title="b4">K</font>,<font color="Olive" title="b2">F</font>) holds <br/><font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">a</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b6">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">c</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">P</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c3">Sigma</font>; <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">F</font> being    <a href="kolmog01.html#M1" title="KOLMOG01:mode.1">ManySortedSigmaField</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">Sigma</font><br/>  for <font color="Olive" title="b2">J</font>, <font color="Olive" title="b3">K</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Olive" title="b2">J</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Olive" title="b3">K</font> holds <br/> for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">c</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font>  st <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Olive" title="b2">J</font>,<font color="Olive" title="b1">F</font>) &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Olive" title="b3">K</font>,<font color="Olive" title="b1">F</font>) holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">c</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be    <a href="kolmog01.html#M1" title="KOLMOG01:mode.1">ManySortedSigmaField</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">Sigma</font>; <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">J</font>, <font color="Olive" title="b2">K</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>  st <font color="Maroon" title="c5">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Olive" title="b1">J</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Olive" title="b2">K</font> holds <br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">c</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Olive" title="b1">J</font>,<font color="Maroon" title="c5">F</font>) &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Olive" title="b2">K</font>,<font color="Maroon" title="c5">F</font>) holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">c</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">J</font>, <font color="Maroon" title="c7">K</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</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="c5">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c6">J</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c7">K</font> implies  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c5">F</font>) &amp; <font color="Olive" title="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c5">F</font>) holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">c</font></span>)</span> )</span><br/>











<span class="kw">assume </span><a NAME="E1:25"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="kolmog01.html#R2" title="KOLMOG01:pred.2">is_independent_wrt</a> <font color="Maroon" title="c4">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">J</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c7">K</font> or  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c5">F</font>) &amp; <font color="Olive" title="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c5">F</font>) holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">c</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">Si</font> = <font color="Maroon" title="c3">Sigma</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">assume </span><a NAME="E3:25"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c6">J</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c7">K</font>
 ; <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">a</font>, <font color="Olive" title="b2">c</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c5">F</font>) &amp; <font color="Olive" title="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c5">F</font>) holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">c</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">c</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</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="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c5">F</font>) &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c5">F</font>) implies <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:25"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Maroon" title="c6">J</font>,<font color="Maroon" title="c5">F</font>)
 <span class="kw">and </span><br/><a NAME="E5:25"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> (<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c5">F</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">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">E1</font> being   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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c12">f1</font> being    <a href="kolmog01.html#M2" title="KOLMOG01:mode.2">SigmaSection</a> of <font color="Maroon" title="c11">E1</font>,<font color="Maroon" title="c5">F</font><span class="kw"> such that </span><br/><a NAME="E7:25"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c11">E1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">J</font>
 <span class="kw">and </span><br/><a NAME="E8:25"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="kolmog01.html#D9" target="_self" title="KOLMOG01:def.9">Def9</a></span>;<br/>
<a NAME="E9:25"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c12">f1</font> <a href="kolmog01.html#R1" title="KOLMOG01:pred.1">is_independent_wrt</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="kolmog01.html#D5" target="_self" title="KOLMOG01:def.5">Def5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">E2</font> being   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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c14">f2</font> being    <a href="kolmog01.html#M2" title="KOLMOG01:mode.2">SigmaSection</a> of <font color="Maroon" title="c13">E2</font>,<font color="Maroon" title="c5">F</font><span class="kw"> such that </span><br/><a NAME="E11:25"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c13">E2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">K</font>
 <span class="kw">and </span><br/><a NAME="E12:25"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="kolmog01.html#D9" target="_self" title="KOLMOG01:def.9">Def9</a></span>;<br/>
<a NAME="E13:25"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<font color="Maroon" title="c14">f2</font> <a href="kolmog01.html#R1" title="KOLMOG01:pred.1">is_independent_wrt</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="kolmog01.html#D5" target="_self" title="KOLMOG01:def.5">Def5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">rngf1</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f1</font>, <font color="Maroon" title="c16">rngf2</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">f2</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">f1</font> = <font color="Maroon" title="c12">f1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c11">E1</font>,<font color="Maroon" title="c15">rngf1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">f2</font> = <font color="Maroon" title="c14">f2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">E2</font>,<font color="Maroon" title="c16">rngf2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E18:25"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c13">E2</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">e2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E20:25"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c20">e2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E21:25"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c20">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">m</font>
 <span class="kw">and </span><br/><a NAME="E22:25"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:25"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></span>;<br/>
<a NAME="E23:25"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon" title="c20">e2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:25"><span class="lab"><font color="Green" title="E33">A14</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">e2</font> = <font color="Maroon" title="c20">e2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">m</font></span>)</span>,<font color="Maroon" title="c13">E2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:25"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E22:25"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="funct_2.html#T1" title="FUNCT_2:th.1">FUNCT_2:1</a></span>;<br/>
<a NAME="E25:25"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c21">e2</font> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:25"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<a NAME="E26:25"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">e2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:25"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="funct_2.html#T14" title="FUNCT_2:th.14">FUNCT_2:14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">e2</font> = <font color="Maroon" title="c21">e2</font> as   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c13">E2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:25"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E22:25"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E25:25"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">f2</font> = <font color="Maroon" title="c18">f2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">E2</font>,<font color="Maroon" title="c8">Si</font> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c23">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">fe2</font> = <font color="Maroon" title="c23">f2</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c22">e2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c8">Si</font> ;<br/>
<a NAME="E30:25"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c23">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:25"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E31:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A18</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">fe2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T29" title="FINSEQ_2:th.29">FINSEQ_2:29</a></span>;<br/>
<a NAME="E32:25"/>
<font color="Maroon" title="c13">E2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E33:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A19</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">e2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E35:25"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
<font color="Maroon" title="c11">E1</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c25">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">e1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E37:25"/><span class="lab"><font color="Green" title="E40">A21</font></span>: 
<font color="Maroon" title="c26">e1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E38:25"/><span class="lab"><font color="Green" title="E41">A22</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c26">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c25">n</font>
 <span class="kw">and </span><br/><a NAME="E39:25"/><span class="lab"><font color="Green" title="E42">A23</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:25"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></span>;<br/>
<a NAME="E40:25"/><span class="lab"><font color="Green" title="E43">A24</font></span>: 
<font color="Maroon" title="c26">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:25"><span class="lab"><font color="Green" title="E42">A23</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">e1</font> = <font color="Maroon" title="c26">e1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c25">n</font></span>)</span>,<font color="Maroon" title="c11">E1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:25"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="txt" href="#E39:25"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="ref" href="funct_2.html#T1" title="FUNCT_2:th.1">FUNCT_2:1</a></span>;<br/>
<a NAME="E42:25"/><span class="lab"><font color="Green" title="E44">A25</font></span>: 
<font color="Maroon" title="c27">e1</font> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:25"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<a NAME="E43:25"/><span class="lab"><font color="Green" title="E45">A26</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c27">e1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:25"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="ref" href="funct_2.html#T14" title="FUNCT_2:th.14">FUNCT_2:14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">e1</font> = <font color="Maroon" title="c27">e1</font> as   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c11">E1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:25"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="txt" href="#E39:25"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="txt" href="#E42:25"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c29">f1</font> = <font color="Maroon" title="c17">f1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c11">E1</font>,<font color="Maroon" title="c8">Si</font> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c30">fe1</font> = <font color="Maroon" title="c29">f1</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c28">e1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c8">Si</font> ;<br/>
<a NAME="E47:25"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c28">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c29">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:25"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E48:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A27</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">fe1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">e1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T29" title="FINSEQ_2:th.29">FINSEQ_2:29</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c31">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E50:25"/><span class="lab"><font color="Green" title="E47">A28</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] implies <font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] implies <font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) ) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="partfun1.html#S1" title="PARTFUN1:sch.1">PARTFUN1:sch 1</a>();<br/></span>
<a NAME="E51:25"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">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"> ( <font color="Maroon" title="c32">x</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"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span> implies <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_1"/><span class="lab"><font color="Green" title="E48">A29</font></span>: 
<font color="Maroon" title="c32">x</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"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span>
 ; <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="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:25_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:25"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="txt" href="#E22:25"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E50:25"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>;<br/>
<a NAME="E3:25_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c31">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:25_1"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</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="E52:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A30</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E53:25"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">x</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"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E54:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A31</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E50">A32</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">fe1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">fe2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:25"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E31:25"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:25"><span class="lab"><font color="Green" title="E49">A31</font></span></a>, <a class="txt" href="#E52:25"><span class="lab"><font color="Green" title="E48">A30</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>
;<br/>
<a NAME="E56:25"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">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"> ( <font color="Maroon" title="c32">x</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"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_3"/><span class="lab"><font color="Green" title="E51">A33</font></span>: 
<font color="Maroon" title="c32">x</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"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</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"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c33">x</font> = <font color="Maroon" title="c32">x</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>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_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="E1:25_3_1"/>
( <font color="Maroon" title="c33">x</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="c30">fe1</font> or  not <font color="Maroon" title="c33">x</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="c30">fe1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:25_3_1_1"/><span class="lab"><font color="Green" title="E52">A34</font></span>: 
<font color="Maroon" title="c33">x</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="c30">fe1</font>
 ; <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"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span><br/><div class="add"><a NAME="E2:25_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A35</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">fe1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E3:25_3_1_1"/><span class="lab"><font color="Green" title="E54">A36</font></span>: 
<font color="Maroon" title="c11">E1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E4:25_3_1_1"/><span class="lab"><font color="Green" title="E55">A37</font></span>: 
<font color="Maroon" title="c33">x</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="c28">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_3_1_1"><span class="lab"><font color="Green" title="E52">A34</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><a NAME="E5:25_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">e1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:25"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><a NAME="E6:25_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:25"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E3:25_3_1_1"><span class="lab"><font color="Green" title="E54">A36</font></span></a></span>;<br/><a NAME="E7:25_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A38</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_3_1_1"><span class="lab"><font color="Green" title="E52">A34</font></span></a>, <a class="txt" href="#E2:25_3_1_1"><span class="lab"><font color="Green" title="E53">A35</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/><a NAME="E8:25_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">e1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_3_1_1"><span class="lab"><font color="Green" title="E55">A37</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E9:25_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:25"><span class="lab"><font color="Green" title="E50">A32</font></span></a>, <a class="txt" href="#E1:25_3"><span class="lab"><font color="Green" title="E51">A33</font></span></a>, <a class="txt" href="#E7:25_3_1_1"><span class="lab"><font color="Green" title="E56">A38</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</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="hsNdiv(this)" href="javascript:()" title="25_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:25_3_1_2"/>
 not <font color="Maroon" title="c33">x</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="c30">fe1</font>
 ; <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"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c34">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:25_3_1_2"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
<font color="Maroon" title="c34">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> <font color="Maroon" title="c24">fe2</font>
 <span class="kw">and </span><br/><a NAME="E4:25_3_1_2"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
<font color="Maroon" title="c33">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">fe1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c34">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_3"><span class="lab"><font color="Green" title="E51">A33</font></span></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></span>;<br/><a NAME="E5:25_3_1_2"/><span class="lab"><font color="Green" title="E54">A41</font></span>: 
<font color="Maroon" title="c34">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> <font color="Maroon" title="c22">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_3_1_2"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><a NAME="E6:25_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A42</font></span>: 
<font color="Maroon" title="c22">e2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:25"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><a NAME="E7:25_3_1_2"/>
<font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c13">E2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">J</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c7">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/><a NAME="E8:25_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c13">E2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E9:25_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c13">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E10:25_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">E2</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c11">E1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></span>;<br/><a NAME="E11:25_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A43</font></span>: 
 not <font color="Maroon" title="c22">e2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_3_1_2"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E12:25_3_1_2"/><span class="lab"><font color="Green" title="E57">A44</font></span>: 
<font color="Maroon" title="c13">E2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font> = 
<font color="Maroon" title="c24">fe2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_3_1_2"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E4:25_3_1_2"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c23">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">e2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_3_1_2"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">e2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:25"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E6:25_3_1_2"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="txt" href="#E11:25_3_1_2"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E12:25_3_1_2"><span class="lab"><font color="Green" title="E57">A44</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:25"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E4:25_3_1_2"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E5:25_3_1_2"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:25_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:25"><span class="lab"><font color="Green" title="E50">A32</font></span></a>, <a class="txt" href="#E1:25_3"><span class="lab"><font color="Green" title="E51">A33</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E57:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A45</font></span>: 
<font color="Maroon" title="c30">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">fe2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c22">e2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:25"><span class="lab"><font color="Green" title="E50">A32</font></span></a>, <a class="ref" href="funct_1.html#D11" title="FUNCT_1:def.11">FUNCT_1:def 11</a></span>;<br/>
<a NAME="E58:25"/>
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font> holds <br/><font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">i</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="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font> implies <font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_4"/><span class="lab"><font color="Green" title="E52">A46</font></span>: 
<font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</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="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_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="E1:25_4_1"/>
( <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font> or  not <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:25_4_1_1"/><span class="lab"><font color="Green" title="E53">A47</font></span>: 
<font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</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="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font></span><br/><div class="add"><a NAME="E2:25_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:25"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E1:25_4"><span class="lab"><font color="Green" title="E52">A46</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_4_1_1"/>
<font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_4_1_1"><span class="lab"><font color="Green" title="E53">A47</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</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="hsNdiv(this)" href="javascript:()" title="25_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:25_4_1_2"/>
 not <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</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="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font></span><br/><div class="add"><a NAME="E2:25_4_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> &amp; <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">E2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:25"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E1:25_4"><span class="lab"><font color="Green" title="E52">A46</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_4_1_2"/>
<font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</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><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">h</font> = <font color="Maroon" title="c31">h</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font></span>)</span>,<font color="Maroon" title="c3">Sigma</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:25"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<a NAME="E60:25"/>
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font> holds <br/><font color="Maroon" title="c32">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c33">i</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="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font> implies <font color="Maroon" title="c32">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c33">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_5"/><span class="lab"><font color="Green" title="E52">A48</font></span>: 
<font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</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="c32">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c33">i</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_5_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="E1:25_5_1"/>
( <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font> or  not <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:25_5_1_1"/><span class="lab"><font color="Green" title="E53">A49</font></span>: 
<font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</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="c32">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c33">i</font></span><br/><div class="add"><a NAME="E2:25_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c29">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c33">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="kolmog01.html#D4" target="_self" title="KOLMOG01:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_5_1_1"/>
<font color="Maroon" title="c32">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c33">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:25"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E1:25_5"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="txt" href="#E1:25_5_1_1"><span class="lab"><font color="Green" title="E53">A49</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:25_5_1_2"/><span class="lab"><font color="Green" title="E53">A50</font></span>: 
 not <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">E1</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="c32">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c33">i</font></span><br/><div class="add"><a NAME="E2:25_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">E2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_5"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E3:25_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c33">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="kolmog01.html#D4" target="_self" title="KOLMOG01:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:25_5_1_2"/>
<font color="Maroon" title="c32">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <font color="Maroon" title="c33">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:25"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E1:25_5"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="txt" href="#E1:25_5_1_2"><span class="lab"><font color="Green" title="E53">A50</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c33">h</font> = <font color="Maroon" title="c32">h</font> as    <a href="kolmog01.html#M2" title="KOLMOG01:mode.2">SigmaSection</a> of <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font>,<font color="Maroon" title="c5">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="kolmog01.html#D4" target="_self" title="KOLMOG01:def.4">Def4</a></span>;<br/>
<a NAME="E62:25"/><span class="lab"><font color="Green" title="E52">A51</font></span>: 
<font color="Maroon" title="c33">h</font> <a href="kolmog01.html#R1" title="KOLMOG01:pred.1">is_independent_wrt</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="kolmog01.html#D5" target="_self" title="KOLMOG01:def.5">Def5</a></span>;<br/>
<a NAME="E63:25"/>
<font color="Maroon" title="c11">E1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E64:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A52</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c28">e1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c34">Pfe1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c29">f1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c28">e1</font>, <font color="Maroon" title="c35">Pfe2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">e2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T32" title="FINSEQ_2:th.32">FINSEQ_2:32</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c36">e2</font> = <font color="Maroon" title="c22">e2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:25"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c37">e1</font> = <font color="Maroon" title="c28">e1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:25"><span class="lab"><font color="Green" title="E53">A52</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E68:25"/>
<font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c13">E2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">J</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c7">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/>
<a NAME="E69:25"/><span class="kw">then </span>
<font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c13">E2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E70:25"/><span class="kw">then </span>
<font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c13">E2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E71:25"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c37">e1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c36">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:25"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="txt" href="#E22:25"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c38">e12</font> = <font color="Maroon" title="c37">e1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c36">e2</font> as   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c11">E1</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c13">E2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T91" title="FINSEQ_3:th.91">FINSEQ_3:91</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c39">e1</font> = <font color="Maroon" title="c37">e1</font> as   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c11">E1</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c40">fe1</font> = <font color="Maroon" title="c30">fe1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c8">Si</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c41">e2</font> = <font color="Maroon" title="c36">e2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c13">E2</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c42">fe2</font> = <font color="Maroon" title="c24">fe2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c8">Si</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c43">f1</font> = <font color="Maroon" title="c29">f1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c11">E1</font>,<font color="Maroon" title="c3">Sigma</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c44">f2</font> = <font color="Maroon" title="c23">f2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c13">E2</font>,<font color="Maroon" title="c3">Sigma</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c45">P</font> = <font color="Maroon" title="c4">P</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">Si</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<span class="lab"><font color="Green" title="E54">A53</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c45">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c33">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c38">e12</font> = 
<font color="Maroon" title="c45">P</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c39">e1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c41">e2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">P</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c40">fe1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">P</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c42">fe2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:25"><span class="lab"><font color="Green" title="E51">A45</font></span></a>, <a class="ref" href="finseqop.html#T9" title="FINSEQOP:th.9">FINSEQOP:9</a></span>
;<br/>
<a NAME="E81:25"/><span class="lab"><font color="Green" title="E55">A54</font></span>: 
( <font color="Maroon" title="c45">P</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c40">fe1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">Pfe1</font> &amp; <font color="Maroon" title="c45">P</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c42">fe2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">Pfe2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c46">P</font> = <font color="Maroon" title="c45">P</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">Sigma</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<a NAME="E83:25"/><span class="lab"><font color="Green" title="E56">A55</font></span>: 
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c46">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c43">f1</font></span>)</span> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c39">e1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c43">f1</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c39">e1</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:25"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="txt" href="#E9:25"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="kolmog01.html#D3" target="_self" title="KOLMOG01:def.3">Def3</a></span>;<br/>
<font color="Maroon" title="c46">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">c</font></span>)</span> = 
<font color="Maroon" title="c46">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c43">f1</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c44">f2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E12:25"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="ref" href="setfam_1.html#T9" title="SETFAM_1:th.9">SETFAM_1:9</a></span>
<br/>.= 
<font color="Maroon" title="c46">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c40">fe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c42">fe2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:25"><span class="lab"><font color="Green" title="E45">A26</font></span></a>, <a class="txt" href="#E26:25"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>
<br/>.= 
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c34">Pfe1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c35">Pfe2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:25"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="txt" href="#E57:25"><span class="lab"><font color="Green" title="E51">A45</font></span></a>, <a class="txt" href="#E62:25"><span class="lab"><font color="Green" title="E52">A51</font></span></a>, <a class="txt" href="#E80:25"><span class="lab"><font color="Green" title="E54">A53</font></span></a>, <a class="txt" href="#E81:25"><span class="lab"><font color="Green" title="E55">A54</font></span></a>, <a class="ref" href="kolmog01.html#D3" target="_self" title="KOLMOG01:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c34">Pfe1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c35">Pfe2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T97" title="RVSUM_1:th.97">RVSUM_1:97</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c46">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E12:25"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E23:25"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E13:25"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E43:25"><span class="lab"><font color="Green" title="E45">A26</font></span></a>, <a class="txt" href="#E26:25"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E83:25"><span class="lab"><font color="Green" title="E56">A55</font></span></a>, <a class="ref" href="kolmog01.html#D3" target="_self" title="KOLMOG01:def.3">Def3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E85:25"/>
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c</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/>


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