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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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">  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><br/>

<div><span class="lab"><font color="Green" title="E8">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">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"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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="E1:7_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <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"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E2:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T38" title="NAT_1:th.38">NAT_1:38</a></span>;<br/><a NAME="E3:7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <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="7_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:7_1_1_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</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"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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"><span class="kw">thus </span><span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T52" title="PRE_POLY:th.52">PRE_POLY:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T52" title="PRE_POLY:th.52">PRE_POLY:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</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="7_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:7_1_1_2"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <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"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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"><span class="kw">thus </span><span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T52" title="PRE_POLY:th.52">PRE_POLY:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1_2"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E2:7"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:7"/>
 <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</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/>


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