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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">s</font> being   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Olive" title="b2">s</font> holds <br/><span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">DecIC</a> (<font color="Olive" title="b2">s</font>,<font color="Olive" title="b3">k</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b3">k</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">s</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">s</font> being   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">DecIC</a> (<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">k</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">k</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="b1">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</font> be   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</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">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c3">s</font> holds <br/><span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">DecIC</a> (<font color="Maroon" title="c3">s</font>,<font color="Olive" title="b1">k</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</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="c3">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">k</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"> ( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c3">s</font> implies <span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">DecIC</a> (<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">k</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="c3">s</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:100"/><span class="lab"><font color="Green" title="E35">Z</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c3">s</font>
 ; <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#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">DecIC</a> (<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">k</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="c3">s</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">DecIC</a> (<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">k</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="memstr_0.html#T65" target="_self" title="MEMSTR_0:th.65">Th188</a></span>
<br/>.= 
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c3">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E35">Z</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</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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