<?xml version="1.0"?>
<div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</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#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a> <a href="ami_7.html#V2" title="AMI_7:attr.2">with_non_trivial_ObjectKinds</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">N</font><br/> for <font color="Olive" title="b2">I</font> being  <a href="ami_1.html#NM4" title="AMI_1:NM.4">Instruction</a> of <font color="Olive" title="b1">A</font> holds   <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Olive" title="b2">I</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">A</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#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a> <a href="ami_7.html#V2" title="AMI_7:attr.2">with_non_trivial_ObjectKinds</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="ami_1.html#NM4" title="AMI_1:NM.4">Instruction</a> of <font color="Maroon" title="c2">A</font> holds   <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Olive" title="b1">I</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">I</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">Instruction</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Maroon" title="c3">I</font></span>)</span></span><br/>





<a NAME="E1:21"/>
for <font color="Olive" title="b1">o</font> being  <a href="ami_1.html#NM1" title="AMI_1:NM.1">Object</a> of <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font> iff <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Maroon" title="c3">I</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">o</font> be   <a href="ami_1.html#NM1" title="AMI_1:NM.1">Object</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font> iff <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Maroon" title="c3">I</font></span>)</span> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Maroon" title="c3">I</font></span>)</span> implies <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Maroon" title="c3">I</font></span>)</span></span><br/><a NAME="E2:21_1_1"/><i><font color="Green" title="E11">A2</font></i>: 
 <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="ref" href="ami_7.html#T10" target="_self" title="AMI_7:th.10">Th10</a></i>;<br/><a NAME="E3:21_1_1"/>
 <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T85" title="XBOOLE_1:th.85">XBOOLE_1:85</a></i>;<br/><a NAME="E4:21_1_1"/><b>then </b>
not <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="#E1:21_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E5:21_1_1"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Maroon" title="c3">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:21_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E2:21_1_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:21_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Maroon" title="c3">I</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font></span><br/>

<a NAME="E3:21_1"/><b>then </b><i><font color="Green" title="E11">A4</font></i>: 
not <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Maroon" title="c3">I</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_2"/>
(  not <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K4" title="AMI_7:func.4">Out_U_Inp</a> <font color="Maroon" title="c3">I</font> or <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font> )
 </span><b>by </b><i><a class="txt" href="#E3:21_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1"><b>suppose </b></a><a NAME="E1:21_1_2_1"/><i><font color="Green" title="E12">A5</font></i>: 
not <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K4" title="AMI_7:func.4">Out_U_Inp</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font></span><br/><div class="add"><a NAME="E2:21_1_2_1"/>
 <a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Maroon" title="c3">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_7.html#K4" title="AMI_7:func.4">Out_U_Inp</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="ref" href="ami_7.html#T11" target="_self" title="AMI_7:th.11">Th11</a></i>;<br/><a NAME="E3:21_1_2_1"/><b>then </b>
not <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="#E1:21_1_2_1"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E4:21_1_2_1"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font>
 <b>by </b><i><a class="txt" href="#E2:21_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_2"><b>suppose </b></a><a NAME="E1:21_1_2_2"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font></span><br/><div class="add"><b>hence </b><a NAME="E2:21_1_2_2"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:21"/>
 <a href="ami_7.html#K3" title="AMI_7:func.3">Out_\_Inp</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_7.html#K2" title="AMI_7:func.2">Output</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="ami_7.html#K5" title="AMI_7:func.5">Input</a> <font color="Maroon" title="c3">I</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
