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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font>, <font color="Olive" title="b2">T</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">S</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">T</font> holds <br/>(<span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">T</font></span>)</span>,<font color="Olive" title="b3">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default">(<font color="Olive" title="b2">T</font>,<font color="Olive" title="b3">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">T</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c2">S</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">T</font> holds <br/>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Olive" title="b1">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Olive" title="b1">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">T</font> implies (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:16"/>
<font color="Maroon" title="c2">S</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span></span><br/>

<a NAME="E2:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:16_1"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1_1"/>
(  not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_1_1"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
 not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:16_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A3</font></span>: 
 not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E3:16_1_1_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
 not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="uproots.html#T6" target="_self" title="UPROOTS:th.6">Th8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="uproots.html#T6" target="_self" title="UPROOTS:th.6">Th8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="uproots.html#T6" target="_self" title="UPROOTS:th.6">Th8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_1_2"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:16_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A6</font></span>: 
 not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:16_1_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_2"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c4">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T7" target="_self" title="UPROOTS:th.7">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_2"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="uproots.html#T7" target="_self" title="UPROOTS:th.7">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_2"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="uproots.html#T6" target="_self" title="UPROOTS:th.6">Th8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:16_1_1_3"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:16_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A8</font></span>: 
 not <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:16_1_1_3"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_3"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default">(<span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c4">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T7" target="_self" title="UPROOTS:th.7">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_3"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="uproots.html#T7" target="_self" title="UPROOTS:th.7">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_3"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="uproots.html#T6" target="_self" title="UPROOTS:th.6">Th8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" title="PRE_POLY:def.5">PRE_POLY:def 5</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:16"/>
(<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">T</font></span>)</span>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">n</font>) <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
