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<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">p</font> being   <a href="memstr_0.html#V3" title="MEMSTR_0:attr.3">data-only</a>  <a href="memstr_0.html#NM1" title="MEMSTR_0:NM.1">PartState</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">q</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="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">q</font> iff <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Olive" title="b3">q</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">p</font> being   <a href="memstr_0.html#V3" title="MEMSTR_0:attr.3">data-only</a>  <a href="memstr_0.html#NM1" title="MEMSTR_0:NM.1">PartState</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">q</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">q</font> iff <font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Olive" title="b2">q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="memstr_0.html#V3" title="MEMSTR_0:attr.3">data-only</a>  <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">  for <font color="Olive" title="b1">q</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="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">q</font> iff <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Olive" title="b1">q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">q</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="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">q</font> iff <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c4">q</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">X</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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>;<br/>
<a NAME="E1:23"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<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>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T59" title="RELAT_1:th.59">RELAT_1:59</a></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="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c4">q</font> implies <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">q</font> )</span>
<div class="add"><a NAME="E1:23_1"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<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>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</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"><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="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>;<br/><a NAME="E2:23_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/><a NAME="E3:23_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<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>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="xboole_1.html#T10" title="XBOOLE_1:th.10">XBOOLE_1:10</a></span>;<br/><span class="kw">assume </span><a NAME="E4:23_1"/>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">q</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="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c4">q</font></span><br/><a NAME="E5:23_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<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>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T76" title="RELAT_1:th.76">RELAT_1:76</a></span>;<br/><a NAME="E6:23_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">p</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</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="memstr_0.html#D6" target="_self" title="MEMSTR_0:def.6">Def23</a></span>;<br/><span class="kw">hence </span><a NAME="E7:23_1"/>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E3:23_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</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">thus </span><a NAME="E3:23"/>
( <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c4">q</font> implies <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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>
