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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="scmring1.html#V1" title="SCMRING1:attr.1">good</a>  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</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">i1</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   <a href="amistd_1.html#K2" title="AMISTD_1:func.2">JUMP</a> <span class="p1">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Olive" title="b1">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">i1</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">  <a href="amistd_1.html#K2" title="AMISTD_1:func.2">JUMP</a> <span class="p1">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">X</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">il</font>)</span>)</span> where <font color="Olive" title="b1">il</font> is    <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>  : verum </span> } </span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">il</font>)</span>)</span> where <font color="Olive" title="b1">il</font> is    <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>  : verum </span> } </span>  implies <font color="Maroon" title="c4">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="c2">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; ( <font color="Maroon" title="c4">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="c2">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">il</font>)</span>)</span> where <font color="Olive" title="b1">il</font> is    <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>  : verum </span> } </span>  ) )</span><br/><div><span class="kw">hereby </span> <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="c4">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="c2">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">il</font>)</span>)</span> where <font color="Olive" title="b1">il</font> is    <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>  : verum </span> } </span>  )</span>
<div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">il1</font> = 1 as    <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>  ;<br/><a NAME="E2:38_1_1"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
 <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p1">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c5">il1</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">il</font>)</span>)</span> where <font color="Olive" title="b1">il</font> is    <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>  : verum </span> } </span> 
 ;<br/><span class="kw">assume </span><a NAME="E3:38_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">il</font>)</span>)</span> where <font color="Olive" title="b1">il</font> is    <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>  : verum </span> } </span> 
 ; <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="c4">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="c2">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E4:38_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p1">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c5">il1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:38_1_1"/>
<font color="Maroon" title="c4">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="c2">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmring3.html#T30" target="_self" title="SCMRING3:th.30">Th59</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><span class="kw">assume </span><a NAME="E2:38_1"/>
<font color="Maroon" title="c4">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="c2">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">il</font>)</span>)</span> where <font color="Olive" title="b1">il</font> is    <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>  : verum </span> } </span> </span><br/><a NAME="E3:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div><span class="lab"><font color="Green" title="E30">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">Y</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">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">il</font>)</span>)</span> where <font color="Olive" title="b1">il</font> is    <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>  : verum </span> } </span>  implies <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:38_1_2"/>
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">il</font>)</span>)</span> where <font color="Olive" title="b1">il</font> is    <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>  : verum </span> } </span> 
 ; <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">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">il</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="E3:38_1_2"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p1">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c6">il</font>)
 ;<br/><a NAME="E4:38_1_2"/>
 <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p1">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c6">il</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmring3.html#T30" target="_self" title="SCMRING3:th.30">Th59</a></span>;<br/><span class="kw">hence </span><a NAME="E5:38_1_2"/>
<font color="Maroon" title="c2">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_2"><span class="lab"><font color="Green" title="E31">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><a NAME="E5:38_1"/>
 <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p1">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Maroon" title="c2">i1</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">il</font>)</span>)</span> where <font color="Olive" title="b1">il</font> is    <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>  : verum </span> } </span> 
 ;<br/><span class="kw">hence </span><a NAME="E6:38_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<span class="p3">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span>,<font color="Olive" title="b1">il</font>)</span>)</span> where <font color="Olive" title="b1">il</font> is    <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>  : verum </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E4:38_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1: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>
<span class="kw">hence </span><a NAME="E2:38"/>
 <a href="amistd_1.html#K2" title="AMISTD_1:func.2">JUMP</a> <span class="p1">(<span class="default"><a href="scmring2.html#K8" title="SCMRING2:func.8">goto</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c1">R</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">i1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI: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>
