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<span class="kw">let </span><font color="Maroon" title="c1">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="c1">Omega</font><br/>  for <font color="Olive" title="b2">Prob</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">A</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b1">Sigma</font><br/>  for <font color="Olive" title="b4">n</font> being    <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>  holds <br/> ( (  <a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b3">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Olive" title="b2">Prob</font> implies <font color="Olive" title="b2">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Olive" title="b3">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">n</font> ) &amp; ( <font color="Olive" title="b3">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Olive" title="b2">Prob</font> implies 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Olive" title="b3">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b3">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">n</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Sigma</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">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">Prob</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c2">Sigma</font><br/>  for <font color="Olive" title="b2">A</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font><br/>  for <font color="Olive" title="b3">n</font> being    <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>  holds <br/> ( (  <a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b2">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Olive" title="b1">Prob</font> implies <font color="Olive" title="b1">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b2">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font> ) &amp; ( <font color="Olive" title="b2">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Olive" title="b1">Prob</font> implies 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b2">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">Prob</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c2">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">A</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font><br/>  for <font color="Olive" title="b2">n</font> being    <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>  holds <br/> ( (  <a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b1">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> implies <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> ) &amp; ( <font color="Olive" title="b1">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> implies 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Olive" title="b1">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">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">n</font> being    <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>  holds <br/> ( (  <a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> implies <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> ) &amp; ( <font color="Maroon" title="c4">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> implies 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> implies <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> ) &amp; ( <font color="Maroon" title="c4">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> implies 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> ) )</span><br/>









<span class="kw">thus </span><a NAME="E1:37"/>
(  <a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> implies <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</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">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font> implies 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:37_1"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 <a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</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="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font></span><br/>

<a NAME="E2:37_1"/>
( <span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p2">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font> &amp; <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <span class="p4">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="bor_cant.html#T10" target="_self" title="BOR_CANT:th.10">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:37_1"/>
<font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:37"/>
<font color="Maroon" title="c4">A</font> <a href="bor_cant.html#R1" title="BOR_CANT:pred.1">is_all_independent_wrt</a> <font color="Maroon" title="c3">Prob</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font></span><br/>

<a NAME="E3:37"/><span class="kw">then </span>
( <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" title="PROB_3:func.1">Partial_Intersection</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bor_cant.html#T10" target="_self" title="BOR_CANT:th.10">Th10</a>, <a class="ref" href="bor_cant.html#T8" target="_self" title="BOR_CANT:th.8">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:37"/>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="series_3.html#K1" title="SERIES_3:func.1">Partial_Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">Prob</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 ; <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>
