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<span class="kw">set </span><font color="Maroon" title="c4">s</font> =  the    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c3">ds</font>,<font color="Maroon" title="c1">D</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">s1</font> =  the    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c3">ds</font>,<font color="Maroon" title="c1">D</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</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="c2">il</font>,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>)</span>)</span>;<br/>
<a NAME="E1:3_1_1"/>
 for <font color="Olive" title="b1">k</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 <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">D</font> holds <br/><span class="p1">(<span class="default"> the    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c3">ds</font>,<font color="Maroon" title="c1">D</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c2">il</font>,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>)</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <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> ; <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="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">D</font> implies <span class="p1">(<span class="default"> the    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c3">ds</font>,<font color="Maroon" title="c1">D</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c2">il</font>,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>)</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:3_1_1_1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">D</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"> the    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c3">ds</font>,<font color="Maroon" title="c1">D</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c2">il</font>,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>)</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E2:3_1_1_1"/><span class="lab"><font color="Green" title="E3">A2</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="c2">il</font>,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</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="tarski.html#K1" title="TARSKI:func.1">}</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/>
<a NAME="E3:3_1_1_1"/>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_3.html#T13" title="AMI_3:th.13">AMI_3:13</a></span>;<br/>
<a NAME="E4:3_1_1_1"/><span class="kw">then </span>
 not  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span> <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="c2">il</font>,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_1_1"><span class="lab"><font color="Green" title="E3">A2</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><span class="p1">(<span class="default"> the    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c3">ds</font>,<font color="Maroon" title="c1">D</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">Start-At</a> (<font color="Maroon" title="c2">il</font>,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>)</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> = 
 the    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c3">ds</font>,<font color="Maroon" title="c1">D</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>
<br/>.= 
<font color="Maroon" title="c1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1_1_1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="scm_1.html#D1" target="_self" title="SCM_1:def.1">Def1</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">s1</font> =  the    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c3">ds</font>,<font color="Maroon" title="c1">D</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</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="c2">il</font>,<a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>)</span>)</span> as    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c3">ds</font>,<font color="Maroon" title="c1">D</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="scm_1.html#D1" target="_self" title="SCM_1:def.1">Def1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">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="c6">s1</font> is <font color="Maroon" title="c2">il</font> <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:3_1_1"/>
 <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">il</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="memstr_0.html#T16" title="MEMSTR_0:th.16">MEMSTR_0:16</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="memstr_0.html#D9" title="MEMSTR_0:def.9">MEMSTR_0:def 9</a> <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>
