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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1</span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1</span>)</span>;<br/>
<a NAME="E1:23_1_1"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">Seq0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:23_1_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c5">Seq0</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c6">Seq0</font> = <font color="Maroon" title="c5">Seq0</font> as   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E5:23_1_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c6">Seq0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">Seq0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<a NAME="E1:23_1_1_1"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E2:23_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">Seq0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_1_1_1"/>
<font color="Maroon" title="c6">Seq0</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">Seq0</font> = <font color="Maroon" title="c6">Seq0</font> as    <a href="measure8.html#M2" title="MEASURE8:mode.2">Set_Sequence</a> of <font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D2" target="_self" title="MEASURE8:def.2">Def2</a></span>;<br/>
<a NAME="E7:23_1_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">Sets</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">Seq0</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_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#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">Sets</font></span>)</span> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">Seq0</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">Sets</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">Seq0</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:23_1_1_2"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E4:23_1_1_2"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">Sets</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:23_1_1_2"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1_2"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="funct_2.html#T190" title="FUNCT_2:th.190">FUNCT_2:190</a></span>;<br/>
<a NAME="E7:23_1_1_2"/>
<font color="Maroon" title="c3">Sets</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c10">n1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c10">n1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D3" target="_self" title="MEASURE8:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">AA</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:23_1_1_2"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">AA</font>
 <span class="kw">and </span><br/><a NAME="E10:23_1_1_2"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c11">AA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c10">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1_2"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E6:23_1_1_2"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E12:23_1_1_2"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c11">AA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <font color="Maroon" title="c10">n1</font></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c12">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:23_1_1_2"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="funct_2.html#T190" title="FUNCT_2:th.190">FUNCT_2:190</a></span>;<br/>
<a NAME="E13:23_1_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="nagata_2.html#K1" title="NAGATA_2:func.1">PairFunc</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a>, <a class="ref" href="nagata_2.html#T2" title="NAGATA_2:th.2">NAGATA_2:2</a></span>;<br/>
<a NAME="E14:23_1_1_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a>  <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"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E15:23_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">n1</font>,<font color="Maroon" title="c12">n2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E17:23_1_1_2"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">n1</font>,<font color="Maroon" title="c12">n2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a>  <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c13">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T190" title="FUNCT_2:th.190">FUNCT_2:190</a></span>;<br/>
<a NAME="E18:23_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">n1</font>,<font color="Maroon" title="c12">n2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:23_1_1_2"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="funct_2.html#T196" title="FUNCT_2:th.196">FUNCT_2:196</a></span>;<br/>
<a NAME="E19:23_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n1</font> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<a NAME="E20:23_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Seq0</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c13">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E9:23_1_1_2"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E12:23_1_1_2"><span class="lab"><font color="Green" title="E22">A8</font></span></a></span>;<br/>
<a NAME="E21:23_1_1_2"/>
<font color="Maroon" title="c7">Seq0</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">Seq0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:23_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">Seq0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:23_1_1_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">Seq0</font> = <font color="Maroon" title="c7">Seq0</font> as    <a href="measure8.html#M3" title="MEASURE8:mode.3">Covering</a> of  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">Sets</font></span>)</span>,<font color="Maroon" title="c2">F</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="measure8.html#D3" target="_self" title="MEASURE8:def.3">Def3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">Seq0</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c8">Seq0</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">Seq0</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span><br/><a NAME="E1:23_1_1_3"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E2:23_1_1_3"/>
<font color="Maroon" title="c8">Seq0</font> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">Cvr</font> <a href="measure8.html#K3" title="MEASURE8:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_2.html#K4" title="FUNCT_2:func.4">pr1</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="measure8.html#K1" title="MEASURE8:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K5" title="FUNCT_2:func.5">pr2</a> <a href="measure8.html#K7" title="MEASURE8:func.7">InvPairFunc</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>


</div>
