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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  1 <a href="struct_0.html#V13" title="STRUCT_0:attr.13">-element</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">s1</font>, <font color="Olive" title="b3">s2</font> being   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <font color="Olive" title="b1">S</font>  st  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Olive" title="b3">s2</font> holds <br/><font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">T</font> be  1 <a href="struct_0.html#V13" title="STRUCT_0:attr.13">-element</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">s1</font>, <font color="Olive" title="b2">s2</font> being   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <font color="Maroon" title="c2">T</font>  st  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Olive" title="b2">s2</font> holds <br/><font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">s2</font> be   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <font color="Maroon" title="c2">T</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#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c4">s2</font> implies <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:14"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c4">s2</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">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font></span><br/>

<a NAME="E2:14"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">s1</font> <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">T</font>
 
<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/>
<a NAME="E3:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">B2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">s2</font>
 
<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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">now </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> <font color="Maroon" title="c3">s1</font> implies <font color="Maroon" title="c3">s1</font> <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">s2</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="E1:14_1"/><span class="lab"><font color="Green" title="E6">A4</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> <font color="Maroon" title="c3">s1</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">s1</font> <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">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/><a NAME="E2:14_1"/><span class="lab"><font color="Green" title="E7">A10</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c3">s1</font> <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> <font color="Maroon" title="c3">s1</font>

<br/>.= 
<font color="Maroon" title="c4">s2</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="#E1:14"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:14_1"><span class="lab"><font color="Green" title="E7">A10</font></span></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="E5:14"/>
<font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E5">B2</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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