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<span class="kw">let </span><font color="Maroon" title="c2">N</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">T</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</a>   <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a>   <a href="ami_1.html#V8" title="AMI_1:attr.8">definite</a>   <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">standard</a>   <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c2">N</font><br/> for <font color="Olive" title="b2">F</font>, <font color="Olive" title="b3">G</font> being  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of   st <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">G</font> holds <br/> <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Olive" title="b2">F</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a>  <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Olive" title="b3">G</font>,<font color="Olive" title="b1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</a>   <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a>   <a href="ami_1.html#V8" title="AMI_1:attr.8">definite</a>   <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">standard</a>   <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c2">N</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">F</font>, <font color="Olive" title="b2">G</font> being  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of   st <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">G</font> holds <br/> <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Olive" title="b1">F</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a>  <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Olive" title="b2">G</font>,<font color="Maroon" title="c3">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font>, <font color="Maroon" title="c5">G</font> be   <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of ; <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="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> implies  <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c4">F</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a>  <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c3">T</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:90"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c4">F</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a>  <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c3">T</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">N</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:90"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<font color="Maroon" title="c6">N</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="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Olive" title="b1">l</font>,<font color="Maroon" title="c3">T</font></span>)</span> where <font color="Olive" title="b1">l</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>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">G</font> </span>}</span> 
 <span class="kw">and </span><br/><a NAME="E4:90"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
 <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">T</font>,<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c6">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="amistd_1.html#D21" target="_self" title="AMISTD_1:def.21">Def21</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">M</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:90"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c7">M</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="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Olive" title="b1">l</font>,<font color="Maroon" title="c3">T</font></span>)</span> where <font color="Olive" title="b1">l</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>  : <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> </span>}</span> 
 <span class="kw">and </span><br/><a NAME="E7:90"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
 <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">T</font>,<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c7">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="amistd_1.html#D21" target="_self" title="AMISTD_1:def.21">Def21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">MM</font> = <font color="Maroon" title="c7">M</font>, <font color="Maroon" title="c9">NN</font> = <font color="Maroon" title="c6">N</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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 <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="membered.html#T3" title="MEMBERED:th.3">MEMBERED:3</a></span>;<br/>
<a NAME="E9:90"/>
<font color="Maroon" title="c7">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">a</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="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font> or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:90_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 ; <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="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N</font></span><br/>

<a NAME="E2:90_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A6</font></span>: 
 ex <font color="Olive" title="b1">l</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 <br/>( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Olive" title="b1">l</font>,<font color="Maroon" title="c3">T</font> &amp; <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E45">A4</font></span></a></span>;<br/>
<a NAME="E3:90_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:90_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E2:90_1"><span class="lab"><font color="Green" title="E47">A6</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>
<a NAME="E10:90"/><span class="kw">then </span>
 <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c8">MM</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c9">NN</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T59" title="XXREAL_2:th.59">XXREAL_2:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:90"/>
 <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c4">F</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a>  <a href="amistd_1.html#K10" title="AMISTD_1:func.10">LastLoc</a> <font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="amistd_1.html#T28" target="_self" title="AMISTD_1:th.28">Th28</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>
