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

<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   <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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="memstr_0.html#V2" title="MEMSTR_0:attr.2">IC-Ins-separated</a>   <a href="memstr_0.html#L1" title="MEMSTR_0:struct.1">Mem-Struct</a> of <font color="Maroon" title="c1">N</font><br/>  for <font color="Olive" title="b2">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b3">p</font> being   <a href="memstr_0.html#NM1" title="MEMSTR_0:NM.1">PartState</a> of <font color="Olive" title="b1">S</font> holds <br/> ( <font color="Olive" title="b3">p</font> is <font color="Olive" title="b2">l</font> <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  iff  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Olive" title="b2">l</font>,<font color="Olive" title="b1">S</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="memstr_0.html#V2" title="MEMSTR_0:attr.2">IC-Ins-separated</a>   <a href="memstr_0.html#L1" title="MEMSTR_0:struct.1">Mem-Struct</a> of <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">  for <font color="Olive" title="b1">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">p</font> being   <a href="memstr_0.html#NM1" title="MEMSTR_0:NM.1">PartState</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <font color="Olive" title="b2">p</font> is <font color="Olive" title="b1">l</font> <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  iff  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Olive" title="b1">l</font>,<font color="Maroon" title="c2">S</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">l</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="memstr_0.html#NM1" title="MEMSTR_0:NM.1">PartState</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <font color="Olive" title="b1">p</font> is <font color="Maroon" title="c3">l</font> <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  iff  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="memstr_0.html#NM1" title="MEMSTR_0:NM.1">PartState</a> of <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"> ( <font color="Maroon" title="c4">p</font> is <font color="Maroon" title="c3">l</font> <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  iff  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">p</font> )</span><br/>







<a NAME="E1:52"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a> </span>)</span></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="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:52"/>
( <font color="Maroon" title="c4">p</font> is <font color="Maroon" title="c3">l</font> <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  implies  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">p</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">p</font> implies <font color="Maroon" title="c4">p</font> is <font color="Maroon" title="c3">l</font> <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:52_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c4">p</font> is <font color="Maroon" title="c3">l</font> <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">p</font></span><br/>

<a NAME="E2:52_1"/>
 <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a>  <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">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="memstr_0.html#D8" target="_self" title="MEMSTR_0:def.8">Def16</a></span>;<br/>
<a NAME="E3:52_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span> <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="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E4:52_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="c5">x</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> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:52_1_1"/><span class="lab"><font color="Green" title="E26">Z</font></span>: 
<font color="Maroon" title="c5">x</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> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</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="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/>

<span class="kw">hence </span><span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>
<br/>.= 
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c4">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="memstr_0.html#D8" target="_self" title="MEMSTR_0:def.8">Def16</a></span>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E26">Z</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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>
<span class="kw">hence </span><a NAME="E5:52_1"/>
 <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_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/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:52"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
 <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">p</font>
 ; <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">p</font> is <font color="Maroon" title="c3">l</font> <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a> </span><br/>

<a NAME="E4:52"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span> <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="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T11" title="RELAT_1:th.11">RELAT_1:11</a></span>;<br/>
<a NAME="E5:52"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
 <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:52"/>
 <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a>  <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">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52"><span class="lab"><font color="Green" title="E25">A6</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="memstr_0.html#D8" target="_self" title="MEMSTR_0:def.8">MEMSTR_0:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">l</font></span><br/>

<span class="kw">thus </span> <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c4">p</font> = 
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c3">l</font>,<font color="Maroon" title="c2">S</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c3">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</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>
