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

<span class="kw">let </span><font color="Maroon" title="c3">e</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">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p2">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span></span>)</span> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:49_2"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p2">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span></span>)</span> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</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="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">fs</font> = <font color="Maroon" title="c3">e</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D7" onmouseover="rs('afinsq_1/D7')" onmouseout="rh()">AFINSQ_1:def 7</a></span>;<br/>
<a NAME="E3:49_2"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
( (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> ) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b2">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/><font color="Olive" title="b2">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">p</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Olive" title="b2">fs1</font></span>)</span> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E23">A10</font></span></a></span>;<br/>
<a NAME="E4:49_2"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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 href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E5:49_2"/><span class="kw">then </span>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E6:49_2"/><span class="kw">then </span>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<a NAME="E7:49_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A14</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <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="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_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="E1:49_2_1"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_2"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:49_2_1_1"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</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="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:49_2_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E3:49_2_1_1"/>
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E3:49_2"><span class="lab"><font color="Green" title="E26">A13</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="49_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:49_2_1_2"/>
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.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"> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:49_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A16</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E23">A10</font></span></a></span>;<br/><a NAME="E3:49_2_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">xx</font> = <font color="Maroon" title="c5">x</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:49_2_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span></span><br/>

<a NAME="E3:49_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">xx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T10" onmouseover="rs('xboole_1/T10')" onmouseout="rh()">XBOOLE_1:10</a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/>
<a NAME="E4:49_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A17</font></span>: 
<font color="Maroon" title="c6">xx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T78" onmouseover="rs('zfmisc_1/T78')" onmouseout="rh()">ZFMISC_1:78</a></span>;<br/>
<a NAME="E5:49_2_1_2_1"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
 <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T2" onmouseover="rs('classes1/T2')" onmouseout="rh()">CLASSES1:2</a></span>;<br/>
<a NAME="E6:49_2_1_2_1"/><span class="lab"><font color="Green" title="E31">A19</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T52" onmouseover="rs('classes1/T52')" onmouseout="rh()">CLASSES1:52</a>, <a class="ref" href="zfmisc_1.html#T77" onmouseover="rs('zfmisc_1/T77')" onmouseout="rh()">ZFMISC_1:77</a></span>;<br/>
<a NAME="E7:49_2_1_2_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T48" onmouseover="rs('classes1/T48')" onmouseout="rh()">CLASSES1:48</a>, <a class="ref" href="classes1.html#T51" onmouseover="rs('classes1/T51')" onmouseout="rh()">CLASSES1:51</a></span>;<br/>
<a NAME="E8:49_2_1_2_1"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_2_1_2_1"><span class="lab"><font color="Green" title="E31">A19</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:49_2_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_2_1_2_1"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E4:49_2_1_2_1"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="classes1.html#T3" onmouseover="rs('classes1/T3')" onmouseout="rh()">CLASSES1:3</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</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="E4:49_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 ;<br/><a NAME="E5:49_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D1" onmouseover="rs('yellow_6/D1')" onmouseout="rh()">YELLOW_6:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:49_2_1_2"/>
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_2_1_2"><span class="lab"><font color="Green" title="E28">A16</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="49_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:49_2_1_3"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">n</font> =  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E3:49_2_1_3"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/><font color="Maroon" title="c6">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c6">fs1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2_1_3"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="txt" href="#E1:49_2"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E23">A10</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">n9</font> = <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E5:49_2_1_3"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <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="#E1:49_2_1_3"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E6:49_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A22</font></span>: 
<font color="Maroon" title="c7">n9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/><a NAME="E7:49_2_1_3"/><span class="lab"><font color="Green" title="E31">A23</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/><font color="Maroon" title="c6">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">p</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" title="c8">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 implies <font color="Maroon" title="c6">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:49_2_1_3_1"/><span class="lab"><font color="Green" title="E32">A24</font></span>: 
( <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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"> <font color="Maroon" title="c6">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span><br/>

<a NAME="E2:49_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A25</font></span>: 
<font color="Maroon" title="c6">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_2_1_3"><span class="lab"><font color="Green" title="E29">A21</font></span></a></span>;<br/>
<a NAME="E3:49_2_1_3_1"/><span class="lab"><font color="Green" title="E34">A26</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_2_1_3"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E1:49_2_1_3_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:49_2_1_3_1"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 &amp;  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49_2_1_3_1"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<a NAME="E5:49_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A27</font></span>: 
( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E6:49_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A28</font></span>: 
( <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 &amp; <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E7:49_2_1_3_1"/><span class="lab"><font color="Green" title="E37">A29</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_2_1_3_1"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E8:49_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A30</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_2_1_3_1"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:49_2_1_3_1"/><span class="lab"><font color="Green" title="E39">A31</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n9</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="afinsq_1.html#T3" onmouseover="rs('afinsq_1/T3')" onmouseout="rh()">AFINSQ_1:3</a></span>;<br/>
<a NAME="E10:49_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A32</font></span>: 
<font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">fs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_2_1_3_1"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="txt" href="#E6:49_2_1_3"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E11:49_2_1_3_1"/>
<font color="Maroon" title="c4">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">fs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_2_1_3_1"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="txt" href="#E9:49_2_1_3_1"><span class="lab"><font color="Green" title="E39">A31</font></span></a>, <a class="txt" href="#E6:49_2_1_3"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E8:49_2_1_3_1"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:49_2_1_3_1"/>
<font color="Maroon" title="c6">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_2_1_3_1"><span class="lab"><font color="Green" title="E33">A25</font></span></a>, <a class="txt" href="#E10:49_2_1_3_1"><span class="lab"><font color="Green" title="E40">A32</font></span></a>, <a class="ref" href="algstr_4.html#T1" target="_self" onmouseover="rs('algstr_4/T1')" onmouseout="rh()">Th1</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="E8:49_2_1_3"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c6">fs1</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_2_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c6">fs1</font></span>)</span> implies <font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:49_2_1_3_2"/>
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c6">fs1</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="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:49_2_1_3_2"/><span class="lab"><font color="Green" title="E32">A33</font></span>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c6">fs1</font></span>)</span> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c6">fs1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E4:49_2_1_3_2"/><span class="lab"><font color="Green" title="E33">A34</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">fs1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_2_1_3_2"><span class="lab"><font color="Green" title="E32">A33</font></span></a>, <a class="ref" href="card_3.html#D3" onmouseover="rs('card_3/D3')" onmouseout="rh()">CARD_3:def 3</a></span>;<br/>
<a NAME="E5:49_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A35</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_2_1_3_2"><span class="lab"><font color="Green" title="E32">A33</font></span></a>, <a class="ref" href="card_3.html#D3" onmouseover="rs('card_3/D3')" onmouseout="rh()">CARD_3:def 3</a></span>;<br/>
<a NAME="E6:49_2_1_3_2"/><span class="lab"><font color="Green" title="E35">A36</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_2_1_3"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="txt" href="#E6:49_2_1_3"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E4:49_2_1_3_2"><span class="lab"><font color="Green" title="E33">A34</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">p</font> = <font color="Maroon" title="c9">p</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49_2_1_3_2"><span class="lab"><font color="Green" title="E33">A34</font></span></a></span>;<br/>
<a NAME="E8:49_2_1_3_2"/>
( <font color="Maroon" title="c10">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c10">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">fs</font></span>)</span> <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="#E6:49_2_1_3"><span class="lab"><font color="Green" title="E30">A22</font></span></a>, <a class="txt" href="#E6:49_2_1_3_2"><span class="lab"><font color="Green" title="E35">A36</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:49_2_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A37</font></span>: 
<font color="Maroon" title="c6">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_2_1_3"><span class="lab"><font color="Green" title="E31">A23</font></span></a></span>;<br/>
<a NAME="E10:49_2_1_3_2"/><span class="lab"><font color="Green" title="E37">A38</font></span>: 
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_2_1_3_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c11">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c11">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:49_2_1_3_2_1"/>
<font color="Maroon" title="c11">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">p</font></span><a href="tarski.html#K1" title="TARSKI:func.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"> <font color="Maroon" title="c11">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> </span><br/>

<a NAME="E2:49_2_1_3_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font>
 
<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="E3:49_2_1_3_2_1"/>
<font color="Maroon" title="c11">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 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>
<a NAME="E11:49_2_1_3_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="49_2_1_3_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">xx</font> = <font color="Maroon" title="c11">x</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:49_2_1_3_2_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">p</font></span><a href="tarski.html#K1" title="TARSKI:func.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"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span></span><br/>

<a NAME="E3:49_2_1_3_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:49_2_1_3_2"><span class="lab"><font color="Green" title="E37">A38</font></span></a></span>;<br/>
<a NAME="E4:49_2_1_3_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">xx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T10" onmouseover="rs('xboole_1/T10')" onmouseout="rh()">XBOOLE_1:10</a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/>
<a NAME="E5:49_2_1_3_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A39</font></span>: 
<font color="Maroon" title="c12">xx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T78" onmouseover="rs('zfmisc_1/T78')" onmouseout="rh()">ZFMISC_1:78</a></span>;<br/>
<a NAME="E6:49_2_1_3_2_2"/><span class="lab"><font color="Green" title="E39">A40</font></span>: 
 <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T2" onmouseover="rs('classes1/T2')" onmouseout="rh()">CLASSES1:2</a></span>;<br/>
<a NAME="E7:49_2_1_3_2_2"/><span class="lab"><font color="Green" title="E40">A41</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T52" onmouseover="rs('classes1/T52')" onmouseout="rh()">CLASSES1:52</a>, <a class="ref" href="zfmisc_1.html#T77" onmouseover="rs('zfmisc_1/T77')" onmouseout="rh()">ZFMISC_1:77</a></span>;<br/>
<a NAME="E8:49_2_1_3_2_2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes1.html#T48" onmouseover="rs('classes1/T48')" onmouseout="rh()">CLASSES1:48</a>, <a class="ref" href="classes1.html#T51" onmouseover="rs('classes1/T51')" onmouseout="rh()">CLASSES1:51</a></span>;<br/>
<a NAME="E9:49_2_1_3_2_2"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_2_1_3_2_2"><span class="lab"><font color="Green" title="E40">A41</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:49_2_1_3_2_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_2_1_3_2_2"><span class="lab"><font color="Green" title="E39">A40</font></span></a>, <a class="txt" href="#E5:49_2_1_3_2_2"><span class="lab"><font color="Green" title="E38">A39</font></span></a>, <a class="ref" href="classes1.html#T3" onmouseover="rs('classes1/T3')" onmouseout="rh()">CLASSES1:3</a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</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="E12:49_2_1_3_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 
;<br/>
<a NAME="E13:49_2_1_3_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D1" onmouseover="rs('yellow_6/D1')" onmouseout="rh()">YELLOW_6:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:49_2_1_3_2"/>
<font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_2_1_3_2"><span class="lab"><font color="Green" title="E34">A35</font></span></a>, <a class="txt" href="#E9:49_2_1_3_2"><span class="lab"><font color="Green" title="E36">A37</font></span></a>, <a class="ref" href="algstr_4.html#T1" target="_self" onmouseover="rs('algstr_4/T1')" onmouseout="rh()">Th1</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="E9:49_2_1_3"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c6">fs1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T76" onmouseover="rs('zfmisc_1/T76')" onmouseout="rh()">ZFMISC_1:76</a></span>;<br/><a NAME="E10:49_2_1_3"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c6">fs1</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E11:49_2_1_3"/>
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K1" title="YELLOW_6:func.1">the_universe_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_2_1_3"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</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>
