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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2">void</a> <a href="ami_1.html#V5">IC-Ins-separated</a> <a href="ami_1.html#V8">definite</a> <a href="ami_7.html#V2" target="_self">with_non_trivial_ObjectKinds</a>  <a href="ami_1.html#L1">AMI-Struct</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ami_1.html#NM3">Instruction</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<a NAME="E1:23"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ami_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_7.html#K4" target="_self">Out_U_Inp</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" target="_self">Output</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" target="_self">Input</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ami_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_7.html#K4" target="_self">Out_U_Inp</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:23_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_7.html#K5" target="_self">Input</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_7.html#K2" target="_self">Output</a> <font color="Maroon">c<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green">E18</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_7.html#K5" target="_self">Input</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_1"/>
(  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_7.html#K4" target="_self">Out_U_Inp</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_7.html#K3" target="_self">Out_\_Inp</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="ami_7.html#E1:23_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1_1"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_7.html#K4" target="_self">Out_U_Inp</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_7.html#K2" target="_self">Output</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ami_7.html#E1:23_1"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_7.html#K3" target="_self">Out_\_Inp</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_2"/>
 <a href="ami_7.html#K3" target="_self">Out_\_Inp</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ami_7.html#K2" target="_self">Output</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="ami_7.html#T10" target="_self">Th10</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_7.html#K2" target="_self">Output</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ami_7.html#E1:23_1_1_1_2"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:23_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" target="_self">Output</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" target="_self">Input</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:23"/>
 <a href="ami_7.html#K4" target="_self">Out_U_Inp</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" target="_self">Output</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" target="_self">Input</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E3:23"/>
(  <a href="ami_7.html#K2" target="_self">Output</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ami_7.html#K4" target="_self">Out_U_Inp</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="ami_7.html#K5" target="_self">Input</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ami_7.html#K4" target="_self">Out_U_Inp</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="ami_7.html#T11" target="_self">Th11</a>, <a class="ref" href="ami_7.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E4:23"/>
<span class="p1">(<span class="default"><a href="ami_7.html#K2" target="_self">Output</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" target="_self">Input</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="ami_7.html#K4" target="_self">Out_U_Inp</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


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