<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">mym</font>, <font color="Maroon" title="c2">myp</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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" title="c1">mym</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">myp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 implies <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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:32"/><span class="lab"><font color="Green" title="E16">A</font></span>: 
( <font color="Maroon" title="c1">mym</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">myp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 ; <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"> <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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  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_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p2">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span> iff <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )</span><br/>

<a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E17">B1</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_1"/><span class="lab"><font color="Green" title="E18">D1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_1_1"/>
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_1"><span class="lab"><font color="Green" title="E18">D1</font></span></a>, <a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">k2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:32_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E19">E1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">E1</font></span></a>, <a class="ref" href="finance2.html#D3" target="_self" onmouseover="rs('finance2/D3')" onmouseout="rh()">Def5</a></span>;<br/><a NAME="E3:32_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E4:32_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:32_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E20">H1</font></span>: 
<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="nat_1.html#T6" onmouseover="rs('nat_1/T6')" onmouseout="rh()">NAT_1:6</a></span>;<br/><a NAME="E4:32_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">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="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">E1</font></span></a>, <a class="txt" href="#E3:32_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E20">H1</font></span></a>, <a class="ref" href="finance2.html#D3" target="_self" onmouseover="rs('finance2/D3')" onmouseout="rh()">Def5</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_1_1_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E21">H1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><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>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  );<br/><a NAME="E2:32_1_1_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_1_1_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E22">J0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1_1_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 ; <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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 
<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="E3:32_1_1_1_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:32_1_1_1_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_1_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T18" onmouseover="rs('numbers/T18')" onmouseout="rh()">NUMBERS:18</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="E2:32_1_1_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E23">J1</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="32_1_1_1_1_1_1_1_2_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E24">K1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">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="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:32_1_1_1_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E3:32_1_1_1_1_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">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="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_1_1_2_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_1_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_1_1_2_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_1_1_1_1_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_1_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E24">K1</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_1_1_2_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_1_1_1_1_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:32_1_1_1_1_1_1_1_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_1_1_1_1_1_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:32_1_1_1_1_1_1_1_2_1_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="#E1:32_1_1_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E22">J0</font></span></a>, <a class="txt" href="#E2:32_1_1_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E23">J1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:32_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E21">H1</font></span></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><span class="kw">hence </span><a NAME="E3:32_1_1_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_1_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:32_1_1_1_1_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_1_1_1_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">kh</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E19">H1</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">kh</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="nat_1.html#T6" onmouseover="rs('nat_1/T6')" onmouseout="rh()">NAT_1:6</a></span>;<br/><a NAME="E4:32_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">kh</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_1"><span class="lab"><font color="Green" title="E18">D1</font></span></a>, <a class="txt" href="#E3:32_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E19">H1</font></span></a>, <a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">kh</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E20">J00</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">kh</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><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>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  );<br/><a NAME="E2:32_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E21">J0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</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>
 ; <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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:32_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E22">E1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c6">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c6">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c6">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_1_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E22">E1</font></span></a>, <a class="ref" href="finance2.html#D3" target="_self" onmouseover="rs('finance2/D3')" onmouseout="rh()">Def5</a></span>;<br/><a NAME="E3:32_1_1_1_1_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E4:32_1_1_1_1_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:32_1_1_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c6">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_1_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E23">H1</font></span>: 
<font color="Maroon" title="c6">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="nat_1.html#T6" onmouseover="rs('nat_1/T6')" onmouseout="rh()">NAT_1:6</a></span>;<br/><a NAME="E4:32_1_1_1_1_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">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="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E22">E1</font></span></a>, <a class="txt" href="#E3:32_1_1_1_1_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E23">H1</font></span></a>, <a class="ref" href="finance2.html#D3" target="_self" onmouseover="rs('finance2/D3')" onmouseout="rh()">Def5</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_1_1_2_1_1_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E24">H1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  );<br/><a NAME="E2:32_1_1_1_1_1_2_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E25">J0</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 ; <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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_1_1_2_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 
<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="E3:32_1_1_1_1_1_2_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:32_1_1_1_1_1_2_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_1_1_1_1_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T18" onmouseover="rs('numbers/T18')" onmouseout="rh()">NUMBERS:18</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="E3:32_1_1_1_1_1_2_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E26">J1</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>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</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="32_1_1_1_1_1_2_1_1_1_1_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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>2</sub></font>[<font color="Maroon" title="c8">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_2_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E27">K1</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">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>2</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:32_1_1_1_1_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E3:32_1_1_1_1_1_2_1_1_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">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="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_1_1_2_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_1_1_2_1_1_1_1_2_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_1_1_2_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_1_1_1_1_2_1_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_1_1_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E27">K1</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_1_1_2_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_1_1_1_2_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_1_1_2_1_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:32_1_1_1_1_1_2_1_1_1_1_2_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_1_1_1_1_1_2_1_1_1_1_2_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E4:32_1_1_1_1_1_2_1_1_1_1_2_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>2</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="#E2:32_1_1_1_1_1_2_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E25">J0</font></span></a>, <a class="txt" href="#E3:32_1_1_1_1_1_2_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E26">J1</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E5:32_1_1_1_1_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_1_1_2_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E24">H1</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_1_1_2_1_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:32_1_1_1_1_1_2_1_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_1_1_1_1_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E3:32_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E22">J1</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="32_1_1_1_1_1_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E23">K0000</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:32_1_1_1_1_1_2_1_1_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</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="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</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="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_1_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">K0000</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E24">H1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  );<br/><a NAME="E4:32_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E25">J0</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_2_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</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>
 ; <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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_1_1_2_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</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>
 
<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="E3:32_1_1_1_1_1_2_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:32_1_1_1_1_1_2_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_1_1_1_1_1_2_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T17" onmouseover="rs('numbers/T17')" onmouseout="rh()">NUMBERS:17</a>, <a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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="E5:32_1_1_1_1_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E26">J1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</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>2</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_2_1_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E27">K0</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</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>2</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">o</font> = <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">o2</font> = <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">o3</font> = <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E2:32_1_1_1_1_1_2_1_1_2_1_1_2_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_2_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_2_1_1_2_1_1_2_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_1_1_2_1_1_2_1_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">k</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="c1">mym</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">k</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">k</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_1_1_2_1_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E27">K0</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">OZ</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">o2</font> = <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2, <font color="Maroon" title="c13">o</font> = <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T17" onmouseover="rs('numbers/T17')" onmouseout="rh()">NUMBERS:17</a></span>;<br/><a NAME="E5:32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c13">o</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c12">o2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rat_1.html#D1" onmouseover="rs('rat_1/D1')" onmouseout="rh()">RAT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_1_1_2_1_1_2_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E28">OZ</font></span></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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:32_1_1_1_1_1_2_1_1_2_1_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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="E6:32_1_1_1_1_1_2_1_1_2_1_1_2"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</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" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E4:32_1_1_1_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E25">J0</font></span></a>, <a class="txt" href="#E5:32_1_1_1_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E26">J1</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E7:32_1_1_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E24">H1</font></span></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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:32_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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="E4:32_1_1_1_1_1_2_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="#E2:32_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E21">J0</font></span></a>, <a class="txt" href="#E3:32_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E22">J1</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E5:32_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E20">J00</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">q</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E20">JK0</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><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>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  );<br/><a NAME="E4:32_1_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E21">J0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</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>
 ; <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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_1_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</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>
 
<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="E3:32_1_1_1_1_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:32_1_1_1_1_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_1_1_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A</font></span></a>, <a class="ref" href="numbers.html#T17" onmouseover="rs('numbers/T17')" onmouseout="rh()">NUMBERS:17</a>, <a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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="E5:32_1_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E22">J1</font></span>: 
 for <font color="Olive" title="b1">k</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">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</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="c7">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_2_1_2_2"/><span class="lab"><font color="Green" title="E23">K0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</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="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">o</font> = <font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">o2</font> = <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">o3</font> = <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E2:32_1_1_1_1_1_2_1_2_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_1_1_2_1_2_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_1_1_2_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</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="c1">mym</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_2_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_1_1_2_1_2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_1_1_1_1_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_1_1_2_1_2_2"><span class="lab"><font color="Green" title="E23">K0</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1_1_2_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_1_1_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_1_1_2_1_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">mym</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:32_1_1_1_1_1_2_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">OZ</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">o2</font> = <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2, <font color="Maroon" title="c12">o</font> = <font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T17" onmouseover="rs('numbers/T17')" onmouseout="rh()">NUMBERS:17</a></span>;<br/><a NAME="E5:32_1_1_1_1_1_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c12">o</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c11">o2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rat_1.html#D1" onmouseover="rs('rat_1/D1')" onmouseout="rh()">RAT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:32_1_1_1_1_1_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_1_1_2_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E24">OZ</font></span></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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:32_1_1_1_1_1_2_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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="E6:32_1_1_1_1_1_2_1_2"/>
 for <font color="Olive" title="b1">k</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">k</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="#E4:32_1_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E21">J0</font></span></a>, <a class="txt" href="#E5:32_1_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E22">J1</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E7:32_1_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E20">JK0</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_2"/><span class="lab"><font color="Green" title="E18">D1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_2_1"/>
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_2"><span class="lab"><font color="Green" title="E18">D1</font></span></a>, <a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">k2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:32_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E19">E1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E19">E1</font></span></a>, <a class="ref" href="finance2.html#D3" target="_self" onmouseover="rs('finance2/D3')" onmouseout="rh()">Def5</a></span>;<br/><a NAME="E3:32_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E4:32_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:32_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E20">H1</font></span>: 
<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="nat_1.html#T6" onmouseover="rs('nat_1/T6')" onmouseout="rh()">NAT_1:6</a></span>;<br/><a NAME="E4:32_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">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">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E19">E1</font></span></a>, <a class="txt" href="#E3:32_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E20">H1</font></span></a>, <a class="ref" href="finance2.html#D3" target="_self" onmouseover="rs('finance2/D3')" onmouseout="rh()">Def5</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E21">H1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><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>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  );<br/><a NAME="E2:32_1_1_1_2_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E22">J0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 ; <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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 
<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="E3:32_1_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:32_1_1_1_2_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_1_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T18" onmouseover="rs('numbers/T18')" onmouseout="rh()">NUMBERS:18</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="E3:32_1_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_1_1_1_2_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E23">J1</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="32_1_1_1_2_1_1_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E24">K1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">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="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:32_1_1_1_2_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E3:32_1_1_1_2_1_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">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">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_2_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_1_1_2_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_2_1_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E24">K1</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_2_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:32_1_1_1_2_1_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_1_1_1_2_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:32_1_1_1_2_1_1_1_2_1_1_2"/>
 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="#E2:32_1_1_1_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E22">J0</font></span></a>, <a class="txt" href="#E1:32_1_1_1_2_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">J1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E3:32_1_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_2_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E21">H1</font></span></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><span class="kw">hence </span><a NAME="E4:32_1_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:32_1_1_1_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_1_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2"/>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">kh</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E19">H1</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">kh</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="nat_1.html#T6" onmouseover="rs('nat_1/T6')" onmouseout="rh()">NAT_1:6</a></span>;<br/><a NAME="E4:32_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">kh</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_2"><span class="lab"><font color="Green" title="E18">D1</font></span></a>, <a class="txt" href="#E3:32_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E19">H1</font></span></a>, <a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_2_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">kh</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E20">J00</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">kh</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><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>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  );<br/><a NAME="E2:32_1_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E21">J0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</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>
 ; <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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_2_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:32_1_1_1_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E22">E1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_2_1_2_1_1_1_1"/>
( <font color="Maroon" title="c6">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c6">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c6">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_2_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E22">E1</font></span></a>, <a class="ref" href="finance2.html#D3" target="_self" onmouseover="rs('finance2/D3')" onmouseout="rh()">Def5</a></span>;<br/><a NAME="E3:32_1_1_1_2_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E4:32_1_1_1_2_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:32_1_1_1_2_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c6">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_2_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E23">H1</font></span>: 
<font color="Maroon" title="c6">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="nat_1.html#T6" onmouseover="rs('nat_1/T6')" onmouseout="rh()">NAT_1:6</a></span>;<br/><a NAME="E4:32_1_1_1_2_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">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">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E22">E1</font></span></a>, <a class="txt" href="#E3:32_1_1_1_2_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E23">H1</font></span></a>, <a class="ref" href="finance2.html#D3" target="_self" onmouseover="rs('finance2/D3')" onmouseout="rh()">Def5</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_2_1_2_1_1_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E24">H1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_2_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  );<br/>
<a NAME="E1:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E25">J0</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 ; <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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</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>
 
<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="E3:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T18" onmouseover="rs('numbers/T18')" onmouseout="rh()">NUMBERS:18</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="E2:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E26">J1</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>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</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="32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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>2</sub></font>[<font color="Maroon" title="c8">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E27">K1</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">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>2</sub></font>[<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E3:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">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">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E27">K1</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:32_1_1_1_2_1_2_1_1_1_1_2_1_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>2</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="#E1:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E25">J0</font></span></a>, <a class="txt" href="#E2:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E26">J1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:32_1_1_1_2_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_2_1_2_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E24">H1</font></span></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><span class="kw">hence </span><a NAME="E3:32_1_1_1_2_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_2_1_2_1_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:32_1_1_1_2_1_2_1_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_1_1_1_2_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E3:32_1_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E22">J1</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="32_1_1_1_2_1_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E23">K0000</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:32_1_1_1_2_1_2_1_1_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</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="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</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="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_2_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_2_1_2_1_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_1_1_2_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E23">K0000</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_2_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E24">H1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  );<br/><a NAME="E4:32_1_1_1_2_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E25">J0</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_2_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</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>
 ; <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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_2_1_2_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</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>
 
<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="E3:32_1_1_1_2_1_2_1_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:32_1_1_1_2_1_2_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">HH</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A</font></span></a></span>;<br/>
<a NAME="E5:32_1_1_1_2_1_2_1_1_2_1_1_2_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:32_1_1_1_2_1_2_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_2_1_2_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E26">HH</font></span></a>, <a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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="E5:32_1_1_1_2_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E26">J1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</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>2</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_2_1_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E27">K0</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</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>2</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:32_1_1_1_2_1_2_1_1_2_1_1_2_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_2_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_2_1_1_2_1_1_2_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_2_1_2_1_1_2_1_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">k</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">myp</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_2_1_1_2_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_2_1_2_1_1_2_1_1_2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">k</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_2_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">k</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_1_1_2_1_2_1_1_2_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_2_1_2_1_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E27">K0</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_1_2_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_2_1_2_1_1_2_1_1_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_1_1_1_2_1_2_1_1_2_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A</font></span></a>, <a class="ref" href="rat_1.html#D2" onmouseover="rs('rat_1/D2')" onmouseout="rh()">RAT_1:def 2</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:32_1_1_1_2_1_2_1_1_2_1_1_2_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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="E6:32_1_1_1_2_1_2_1_1_2_1_1_2"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</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" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E4:32_1_1_1_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E25">J0</font></span></a>, <a class="txt" href="#E5:32_1_1_1_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E26">J1</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E7:32_1_1_1_2_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E24">H1</font></span></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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:32_1_1_1_2_1_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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="E4:32_1_1_1_2_1_2_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="#E2:32_1_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E21">J0</font></span></a>, <a class="txt" href="#E3:32_1_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E22">J1</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E5:32_1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E20">J00</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">q</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E20">JK0</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><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>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> $1 implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  );<br/><a NAME="E4:32_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E21">J0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_2_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</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>
 ; <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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_2_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</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>
 
<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="E3:32_1_1_1_2_1_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E4:32_1_1_1_2_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">HH</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A</font></span></a></span>;<br/>
<a NAME="E5:32_1_1_1_2_1_2_1_2_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" onmouseover="rs('int_1/D1')" onmouseout="rh()">INT_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:32_1_1_1_2_1_2_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E22">HH</font></span></a>, <a class="ref" href="numbers.html#T14" onmouseover="rs('numbers/T14')" onmouseout="rh()">NUMBERS:14</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="E5:32_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E22">J1</font></span>: 
 for <font color="Olive" title="b1">k</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">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</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="c7">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_2_1_2_2"/><span class="lab"><font color="Green" title="E23">K0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</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="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">o3</font> = <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E2:32_1_1_1_2_1_2_1_2_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_1_1_2_1_2_1_2_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/>

<a NAME="E2:32_1_1_1_2_1_2_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</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">myp</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_2_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_1_1_2_1_2_1_2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_1_1_2_1_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1_2_1_2_1_2_2"><span class="lab"><font color="Green" title="E23">K0</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_1_1_2_1_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</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"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> </span><br/><div class="add"><a NAME="E2:32_1_1_1_2_1_2_1_2_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">myp</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">kh</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_1_1_1_2_1_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A</font></span></a>, <a class="ref" href="rat_1.html#D2" onmouseover="rs('rat_1/D2')" onmouseout="rh()">RAT_1:def 2</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:32_1_1_1_2_1_2_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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="E6:32_1_1_1_2_1_2_1_2"/>
 for <font color="Olive" title="b1">k</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">k</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="#E4:32_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E21">J0</font></span></a>, <a class="txt" href="#E5:32_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E22">J1</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E7:32_1_1_1_2_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E20">JK0</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:32_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">m</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>, <font color="Maroon" title="c6">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:32_1_2"/><span class="lab"><font color="Green" title="E18">C0</font></span>: 
( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c6">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rat_1.html#T8" onmouseover="rs('rat_1/T8')" onmouseout="rh()">RAT_1:8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">m2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:32_1_2"/><span class="lab"><font color="Green" title="E19">C2</font></span>: 
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m2</font> or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c7">m2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T2" onmouseover="rs('int_1/T2')" onmouseout="rh()">INT_1:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_2_1"/>
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m2</font> or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c7">m2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_2"><span class="lab"><font color="Green" title="E19">C2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_1"/><span class="lab"><font color="Green" title="E20">S1</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m2</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c8">q2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_2_1_1"/><span class="lab"><font color="Green" title="E21">F2</font></span>: 
<font color="Maroon" title="c8">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_2"><span class="lab"><font color="Green" title="E18">C0</font></span></a></span>;<br/><a NAME="E4:32_1_2_1_1"/><span class="lab"><font color="Green" title="E22">G1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_1_2_1_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="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">m2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:32_1_2_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_2"><span class="lab"><font color="Green" title="E18">C0</font></span></a>, <a class="txt" href="#E1:32_1_2_1_1"><span class="lab"><font color="Green" title="E20">S1</font></span></a>, <a class="txt" href="#E3:32_1_2_1_1"><span class="lab"><font color="Green" title="E21">F2</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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="E5:32_1_2_1_1"/><span class="lab"><font color="Green" title="E23">G2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_2_1_1_2_1"/>
( <font color="Maroon" title="c8">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c8">q2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c8">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <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>, <a class="txt" href="#E4:32_1_2_1_1"><span class="lab"><font color="Green" title="E22">G1</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_1_2_1_2"/>
<font color="Maroon" title="c8">q2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">q3</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_2_1_1_2_1_2"/><span class="lab"><font color="Green" title="E24">F2</font></span>: 
<font color="Maroon" title="c9">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E4:32_1_2_1_1_2_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_1_2_1_1_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</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="c9">q3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">q3</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="c1">mym</font>,<font color="Maroon" title="c7">m2</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="c9">q3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<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/>
<span class="kw">hence </span><a NAME="E2:32_1_2_1_1_2_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_2_1_1_2_1_2"><span class="lab"><font color="Green" title="E24">F2</font></span></a>, <a class="txt" href="#E4:32_1_2_1_1"><span class="lab"><font color="Green" title="E22">G1</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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><span class="kw">hence </span><a NAME="E5:32_1_2_1_1_2_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E6:32_1_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_1_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_2_1_1_3_1"/>
( <font color="Maroon" title="c7">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">m2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E24">I1</font></span>: 
<font color="Maroon" title="c7">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font></span><br/><div class="add"><a NAME="E2:32_1_2_1_1_3_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_2_1_1"><span class="lab"><font color="Green" title="E23">G2</font></span></a>, <a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_1_2_1_1_3_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E24">I1</font></span></a>, <a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_1_3_1_2"/>
<font color="Maroon" title="c7">m2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">m3</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_2_1_1_3_1_2"/><span class="lab"><font color="Green" title="E24">F2</font></span>: 
<font color="Maroon" title="c9">m3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E4:32_1_2_1_1_3_1_2"/><span class="lab"><font color="Green" title="E25">I2</font></span>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">m3</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E24">F2</font></span></a>, <a class="txt" href="#E5:32_1_2_1_1"><span class="lab"><font color="Green" title="E23">G2</font></span></a>, <a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><a NAME="E5:32_1_2_1_1_3_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_1_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_1_2_1_1_3_1_2_1"/>
<span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</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="c9">m3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">m3</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c1">mym</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">m3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:32_1_2_1_1_3_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E25">I2</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="txt" href="#E3:32_1_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E24">F2</font></span></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><span class="kw">hence </span><a NAME="E6:32_1_2_1_1_3_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E7:32_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E8:32_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_2"/><span class="lab"><font color="Green" title="E20">S1</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c7">m2</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" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c8">q2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_2_1_2"/><span class="lab"><font color="Green" title="E21">F2</font></span>: 
<font color="Maroon" title="c8">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_2"><span class="lab"><font color="Green" title="E18">C0</font></span></a></span>;<br/><a NAME="E4:32_1_2_1_2"/><span class="lab"><font color="Green" title="E22">G1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_1_2_1_2_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">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">m2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D2" target="_self" onmouseover="rs('finance2/D2')" onmouseout="rh()">Def4</a>, <a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:32_1_2_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K3" title="FINANCE2:func.3">GoCross_Seq_REAL</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_2"><span class="lab"><font color="Green" title="E18">C0</font></span></a>, <a class="txt" href="#E1:32_1_2_1_2"><span class="lab"><font color="Green" title="E20">S1</font></span></a>, <a class="txt" href="#E3:32_1_2_1_2"><span class="lab"><font color="Green" title="E21">F2</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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="E5:32_1_2_1_2"/><span class="lab"><font color="Green" title="E23">G2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_2_1_2_2_1"/>
( <font color="Maroon" title="c8">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c8">q2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_2_2_1_1"/>
<font color="Maroon" title="c8">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:32_1_2_1_2_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <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>, <a class="txt" href="#E4:32_1_2_1_2"><span class="lab"><font color="Green" title="E22">G1</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_2_2_1_2"/>
<font color="Maroon" title="c8">q2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">q3</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_2_1_2_2_1_2"/><span class="lab"><font color="Green" title="E24">F2</font></span>: 
<font color="Maroon" title="c9">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E4:32_1_2_1_2_2_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_2_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_1_2_1_2_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</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="c9">q3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">q3</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">myp</font>,<font color="Maroon" title="c7">m2</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="c9">q3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<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/>
<span class="kw">hence </span><a NAME="E2:32_1_2_1_2_2_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E24">F2</font></span></a>, <a class="txt" href="#E4:32_1_2_1_2"><span class="lab"><font color="Green" title="E22">G1</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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><span class="kw">hence </span><a NAME="E5:32_1_2_1_2_2_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">q2</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E6:32_1_2_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_2_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_2_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:32_1_2_1_2_3_1"/>
( <font color="Maroon" title="c7">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">m2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_2_3_1_1"/>
<font color="Maroon" title="c7">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font></span><br/><div class="add"><a NAME="E2:32_1_2_1_2_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<font color="Maroon" title="c7">m2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_1_2_1_2_3_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_2_1_2"><span class="lab"><font color="Green" title="E23">G2</font></span></a>, <a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:32_1_2_1_2_3_1_2"/>
<font color="Maroon" title="c7">m2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">m3</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:32_1_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E24">F2</font></span>: 
<font color="Maroon" title="c9">m3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">m2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E4:32_1_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E25">I2</font></span>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">m3</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">m3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E24">F2</font></span></a>, <a class="txt" href="#E5:32_1_2_1_2"><span class="lab"><font color="Green" title="E23">G2</font></span></a>, <a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><a NAME="E5:32_1_2_1_2_3_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2_1_2_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:32_1_2_1_2_3_1_2_1"/>
<span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</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="c9">m3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">m3</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K5" title="FINANCE2:func.5">GoCross_Partial_Union</a> (<font color="Maroon" title="c2">myp</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">m3</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finance2.html#D4" target="_self" onmouseover="rs('finance2/D4')" onmouseout="rh()">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:32_1_2_1_2_3_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E25">I2</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="txt" href="#E3:32_1_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E24">F2</font></span></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><span class="kw">hence </span><a NAME="E6:32_1_2_1_2_3_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">m2</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E7:32_1_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T12" onmouseover="rs('prob_1/T12')" onmouseout="rh()">PROB_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E8:32_1_2_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:32_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span> iff <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E17">B1</font></span></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>
<span class="kw">hence </span><a NAME="E3:32"/>
<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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c1">mym</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</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="finance2.html#K7" title="FINANCE2:func.7">GoCross_Union</a> <font color="Maroon" title="c2">myp</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</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"> verum</span><br/>


</div>
