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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">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">B</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="Olive" title="b1">Sigma</font><br/>  for <font color="Olive" title="b4">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b3">B</font><br/>  for <font color="Olive" title="b5">f</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Olive" title="b1">Sigma</font>  st (  for <font color="Olive" title="b6">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> <br/>  for <font color="Olive" title="b7">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b3">B</font> holds  <font color="Olive" title="b2">P</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"><font color="Olive" title="b5">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b6">n</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Olive" title="b7">b</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_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b6">n</font></span>)</span></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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">b</font></span>)</span> ) &amp; <font color="Olive" title="b5">f</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  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="b4">b</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b5">f</font></span>)</span></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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">b</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> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b5">f</font></span>)</span></span>)</span></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">P</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">B</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="c2">Sigma</font><br/>  for <font color="Olive" title="b3">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b2">B</font><br/>  for <font color="Olive" title="b4">f</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>  st (  for <font color="Olive" title="b5">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> <br/>  for <font color="Olive" title="b6">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b2">B</font> holds  <font color="Olive" title="b1">P</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"><font color="Olive" title="b4">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Olive" title="b6">b</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_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b5">n</font></span>)</span></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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">b</font></span>)</span> ) &amp; <font color="Olive" title="b4">f</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  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="b3">b</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b4">f</font></span>)</span></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_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">b</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> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b4">f</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</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">B</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="c2">Sigma</font><br/>  for <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">B</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>  st (  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> <br/>  for <font color="Olive" title="b5">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">B</font> holds  <font color="Maroon" title="c3">P</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"><font color="Olive" title="b3">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Olive" title="b5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">f</font> <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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">b</font></span>)</span> ) &amp; <font color="Olive" title="b3">f</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  holds <br/><font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b3">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</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="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b3">f</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">B</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="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">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">B</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>  st (  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> <br/>  for <font color="Olive" title="b4">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">B</font> holds  <font color="Maroon" title="c3">P</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"><font color="Olive" title="b2">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Olive" title="b4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">b</font></span>)</span> ) &amp; <font color="Olive" title="b2">f</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  holds <br/><font color="Maroon" title="c3">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">b</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</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="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b2">f</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">B</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="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>  st (  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> <br/>  for <font color="Olive" title="b3">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">B</font> holds  <font color="Maroon" title="c3">P</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"><font color="Olive" title="b1">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Olive" title="b3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">b</font></span>)</span> ) &amp; <font color="Olive" title="b1">f</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  holds <br/><font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</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="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">f</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> <br/>  for <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">B</font> holds  <font color="Maroon" title="c3">P</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"><font color="Maroon" title="c6">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span> ) &amp; <font color="Maroon" title="c6">f</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  implies <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</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="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> )</span><br/>











<span class="kw">reconsider </span><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Sigma</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font> as   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<a NAME="E3:5"/>
 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  <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">f</font>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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"> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">f</font>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c9">n</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font></span><br/>

<a NAME="E1:5_1"/>
<font color="Maroon" title="c7">b</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E2:5_1"/>
<span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">f</font>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c9">n</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#D1" title="DYNKIN:def.1">DYNKIN:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">seqIntf</font> =  <a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">f</font>) as   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></span>;<br/>
<a NAME="E5:5"/>
 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  <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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"> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c10">n</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font></span><br/>

<a NAME="E1:5_2"/>
<font color="Maroon" title="c7">b</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E2:5_2"/>
<span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c10">n</font> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#D1" title="DYNKIN:def.1">DYNKIN:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">seqIntPf</font> =  <a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>) as   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></span>;<br/>
<span class="lab"><font color="Green" title="E6">A1</font></span>: <font color="Maroon" title="c7">b</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c6">f</font></span>)</span> = 
<font color="Maroon" title="c7">b</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T15" title="PROB_3:th.15">PROB_3:15</a></span>
<br/>.= 
 <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c10">seqIntPf</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#T10" title="DYNKIN:th.10">DYNKIN:10</a></span>
;<br/>
<span class="kw">assume </span><a NAME="E8:5"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 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> <br/>  for <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">B</font> holds  <font color="Maroon" title="c3">P</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"><font color="Maroon" title="c6">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span>
 ; <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">f</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>  or <font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</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="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font> = 
<font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seqIntf</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<font color="Maroon" title="c3">P</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"><font color="Maroon" title="c6">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c7">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#D1" title="DYNKIN:def.1">DYNKIN:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:5"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">r</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T9" title="SEQ_1:th.9">SEQ_1:9</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="E10:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c9">seqIntf</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c6">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D7" title="FUNCT_2:def.7">FUNCT_2:def 7</a></span>;<br/>
<a NAME="E11:5"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">n</font>, <font color="Olive" title="b3">m</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>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b3">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">m</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">Pf</font> =  <a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font> as   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c13">n</font>, <font color="Maroon" title="c14">m</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"> ( <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">n</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c14">m</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:5_5"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m</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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">n</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c14">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:5_5"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c14">m</font></span><br/>

<a NAME="E4:5_5"/>
<font color="Maroon" title="c12">Pf</font> is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T11" title="PROB_3:th.11">PROB_3:11</a></span>;<br/>
<a NAME="E5:5_5"/><span class="kw">then </span>
<font color="Maroon" title="c12">Pf</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">Pf</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c14">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_5"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="prob_1.html#D5" title="PROB_1:def.5">PROB_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:5_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c14">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_5"><span class="lab"><font color="Green" title="E11">A6</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>
<a NAME="E12:5"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b2">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">n</font>, <font color="Maroon" title="c12">m</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"> ( <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> implies <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">m</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:5_6"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</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"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">m</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:5_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">m</font></span><br/>

<a NAME="E3:5_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#D1" title="DYNKIN:def.1">DYNKIN:def 1</a></span>;<br/>
<a NAME="E4:5_6"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:5_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:5"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E1:5_6"><span class="lab"><font color="Green" title="E10">A7</font></span></a></span>;<br/>
<a NAME="E6:5_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_6"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:5_6"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">b</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:5_6"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:5_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#D1" title="DYNKIN:def.1">DYNKIN:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A10</font></span>: 
 <a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>) is  <a href="prob_1.html#V3" title="PROB_1:attr.3">non-descending</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D5" title="PROB_1:def.5">PROB_1:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E14:5"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</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="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</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="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span><br/>

<a NAME="E15:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A12</font></span>: 
 <a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">f</font>) is  <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#T9" title="DYNKIN:th.9">DYNKIN:9</a></span>;<br/>
<a NAME="E16:5"/>
 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  <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">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"> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font></span><br/>

<a NAME="E1:5_7"/><span class="lab"><font color="Green" title="E13">A13</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_7_1"/><span class="lab"><font color="Green" title="E14">A14</font></span>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</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">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">k</font> = <font color="Maroon" title="c12">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seqIntf</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#D2" title="PROB_3:def.2">PROB_3:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</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="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seqIntf</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_7_1"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="dynkin.html#D1" title="DYNKIN:def.1">DYNKIN:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</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="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#D1" title="DYNKIN:def.1">DYNKIN:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><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="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c13">k</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#D2" title="PROB_3:def.2">PROB_3:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#D1" title="DYNKIN:def.1">DYNKIN:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:5_7_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
<font color="Maroon" title="c9">seqIntf</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#D2" title="PROB_3:def.2">PROB_3:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#D1" title="DYNKIN:def.1">DYNKIN:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#D2" title="PROB_3:def.2">PROB_3:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="dynkin.html#D1" title="DYNKIN:def.1">DYNKIN:def 1</a></span>
;<br/>
<a NAME="E3:5_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A15</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E4:5_7"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E3:5_7"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="txt" href="#E1:5_7"><span class="lab"><font color="Green" title="E13">A13</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:5_7"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c11">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">then </span><font color="Maroon" title="c3">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="dynkin.html#K2" title="DYNKIN:func.2">seqIntersection</a> (<font color="Maroon" title="c7">b</font>,<span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span>)</span>)</span> = 
<font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p1">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>
<br/>.= 
 <a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c9">seqIntf</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:5"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="prob_3.html#T44" title="PROB_3:th.44">PROB_3:44</a></span>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="series_1.html#K2" title="SERIES_1:func.2">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:5"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="series_1.html#T9" title="SERIES_1:th.9">SERIES_1:9</a></span>
<br/>.= 
<font color="Maroon" title="c8">r</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:5"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="ref" href="prob_3.html#T44" title="PROB_3:th.44">PROB_3:44</a></span>
;<br/>
<span class="kw">then </span><font color="Maroon" title="c8">r</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K2" title="PROB_3:func.2">Partial_Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span></span>)</span> = 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K8" title="PROB_1:func.8">*</a> <font color="Maroon" title="c10">seqIntPf</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T41" title="PROB_3:th.41">PROB_3:41</a>, <a class="ref" href="seq_2.html#T8" title="SEQ_2:th.8">SEQ_2:8</a></span>
<br/>.= 
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:5"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E7:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="prob_2.html#T10" title="PROB_2:th.10">PROB_2:10</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E19:5"/>
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</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="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_3.html#T41" title="PROB_3:th.41">PROB_3:41</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>
