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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><a href="finance2.html#K4" title="FINANCE2:func.4">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">pm</font>,<font color="Maroon" title="c1">k</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> ;<br/>
<a NAME="E1:25_1_1"/>
<span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">pm</font>,<font color="Maroon" title="c1">k</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T25" onmouseover="rs('prob_1/T25')" onmouseout="rh()">PROB_1:25</a></span>;<br/>
<a NAME="E2:25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">J1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D3" target="_self" onmouseover="rs('finance2/D3')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E3:25_1_1"/><span class="lab"><font color="Green" title="E15">J2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_1_1_1"/><span class="lab"><font color="Green" title="E16">B1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:25_1_1_1"/><span class="lab"><font color="Green" title="E17">C2</font></span>: 
<span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">pm</font>,<font color="Maroon" title="c1">k</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T25" onmouseover="rs('prob_1/T25')" onmouseout="rh()">PROB_1:25</a></span>;<br/>
<a NAME="E3:25_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K4" title="FINANCE2:func.4">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">pm</font>,<font color="Maroon" title="c1">k</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">pm</font>,<font color="Maroon" title="c1">k</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> is    <a href="prob_1.html#M1" title="PROB_1:mode.1">Event</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1_1_1"><span class="lab"><font color="Green" title="E16">B1</font></span></a>, <a class="txt" href="#E2:25_1_1_1"><span class="lab"><font color="Green" title="E17">C2</font></span></a>, <a class="ref" href="prob_1.html#T21" onmouseover="rs('prob_1/T21')" onmouseout="rh()">PROB_1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:25_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D3" target="_self" onmouseover="rs('finance2/D3')" onmouseout="rh()">Def5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E4:25_1_1"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:25_1_1"><span class="lab"><font color="Green" title="E14">J1</font></span></a>, <a class="txt" href="#E3:25_1_1"><span class="lab"><font color="Green" title="E15">J2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:25_1_1"/>
 <a href="finance2.html#K4" title="FINANCE2:func.4">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">pm</font>,<font color="Maroon" title="c1">k</font>) is   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T25" onmouseover="rs('prob_1/T25')" onmouseout="rh()">PROB_1:25</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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