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<span class="kw">let </span><font color="Maroon" title="c1">N</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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="Red">::  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="ami_1.html#V2" title="AMI_1:attr.2">stored-program</a>   <a href="ami_1.html#V4" title="AMI_1:attr.4">halting</a>   <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a>   <a href="ami_1.html#V7" title="AMI_1:attr.7">steady-programmed</a>   <a href="ami_1.html#V8" title="AMI_1:attr.8">definite</a>   <a href="ami_1.html#V9" title="AMI_1:attr.9">realistic</a>   <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c1">N</font><br/> for <font color="Olive" title="b2">p</font> being  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>  the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b3">d</font> being  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <font color="Olive" title="b1">S</font> holds <br/> ( <font color="Olive" title="b2">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b3">d</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Olive" title="b1">S</font> iff <font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">d</font> <a href="ami_1.html#R2" title="AMI_1:pred.2">computes</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</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="ami_1.html#V2" title="AMI_1:attr.2">stored-program</a>   <a href="ami_1.html#V4" title="AMI_1:attr.4">halting</a>   <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a>   <a href="ami_1.html#V7" title="AMI_1:attr.7">steady-programmed</a>   <a href="ami_1.html#V8" title="AMI_1:attr.8">definite</a>   <a href="ami_1.html#V9" title="AMI_1:attr.9">realistic</a>   <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>  the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Maroon" title="c2">S</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b2">d</font> being  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b2">d</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font> iff <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">d</font> <a href="ami_1.html#R2" title="AMI_1:pred.2">computes</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>  the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Maroon" title="c2">S</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">d</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font> iff <font color="Maroon" title="c3">p</font>,<font color="Olive" title="b1">d</font> <a href="ami_1.html#R2" title="AMI_1:pred.2">computes</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">d</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font> iff <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">d</font> <a href="ami_1.html#R2" title="AMI_1:pred.2">computes</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>







<span class="kw">thus </span><a NAME="E1:137"/>
( <font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">d</font> <a href="ami_1.html#R2" title="AMI_1:pred.2">computes</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">d</font> <a href="ami_1.html#R2" title="AMI_1:pred.2">computes</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="137_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:137_1"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </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#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:137_1"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">d</font> <a href="ami_1.html#R2" title="AMI_1:pred.2">computes</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="ami_1.html#D30" target="_self" title="AMI_1:def.30">AMI_1:def 30</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> implies  ex <font color="Olive" title="b1">s</font> being  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">s</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">s</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:137_1"/>
<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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">s</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">s</font></span>)</span> )</span><br/>

<a NAME="E4:137_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:137_1"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c5">x</font> as   <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T174" title="FUNCT_1:th.174">FUNCT_1:174</a>, <a class="ref" href="relat_1.html#T206" title="RELAT_1:th.206">RELAT_1:206</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">s</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c6">s</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c6">s</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E6:137_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c6">s</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c6">s</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E7:137_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c6">s</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:137_1"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E4:137_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c6">s</font></span>)</span></span><br/>

<a NAME="E8:137_1"/>
<span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:137_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:137_1"/>
<span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c6">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<a NAME="E2:137"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </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#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<a NAME="E3:137"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A4</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:137"/>
<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">d</font> <a href="ami_1.html#R2" title="AMI_1:pred.2">computes</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E5:137"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font> being  <a href="ami_1.html#NM4" title="AMI_1:NM.4">FinPartState</a> of <font color="Maroon" title="c2">S</font> st <br/>(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">s</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font> &amp; <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_1.html#K19" title="AMI_1:func.19">Result</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">s</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:137"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="ami_1.html#D30" target="_self" title="AMI_1:def.30">Def29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:137"/>
<font color="Maroon" title="c3">p</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c4">d</font> is   <a href="ami_1.html#NM6" title="AMI_1:NM.6">pre-program</a> of <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
