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<b>let </b><font color="Maroon" title="c1">IL</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">N</font> being <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> <br/> for <font color="Olive" title="b2">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c1">IL</font>,<font color="Olive" title="b1">N</font><br/> for <font color="Olive" title="b3">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">la</font> being  <a href="ami_1.html#NM1" title="AMI_1:NM.1">Object</a> of <font color="Olive" title="b2">S</font>  st  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Olive" title="b4">la</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> holds <br/>for <font color="Olive" title="b5">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b3">A</font> holds  <font color="Olive" title="b4">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Olive" title="b5">a</font> is   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <font color="Olive" title="b2">S</font></span><br/><b>let </b><font color="Maroon" title="c2">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">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c1">IL</font>,<font color="Maroon" title="c2">N</font><br/> for <font color="Olive" title="b2">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">la</font> being  <a href="ami_1.html#NM1" title="AMI_1:NM.1">Object</a> of <font color="Olive" title="b1">S</font>  st  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Olive" title="b3">la</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> holds <br/>for <font color="Olive" title="b4">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">A</font> holds  <font color="Olive" title="b3">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Olive" title="b4">a</font> is   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <font color="Olive" title="b1">S</font></span><br/><b>let </b><font color="Maroon" title="c3">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c1">IL</font>,<font color="Maroon" title="c2">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">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">la</font> being  <a href="ami_1.html#NM1" title="AMI_1:NM.1">Object</a> of <font color="Maroon" title="c3">S</font>  st  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Olive" title="b2">la</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> holds <br/>for <font color="Olive" title="b3">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">A</font> holds  <font color="Olive" title="b2">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Olive" title="b3">a</font> is   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <font color="Maroon" title="c3">S</font></span><br/><b>let </b><font color="Maroon" title="c4">A</font> be    <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">la</font> being  <a href="ami_1.html#NM1" title="AMI_1:NM.1">Object</a> of <font color="Maroon" title="c3">S</font>  st  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Olive" title="b1">la</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> holds <br/>for <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font> holds  <font color="Olive" title="b1">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Olive" title="b2">a</font> is   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <font color="Maroon" title="c3">S</font></span><br/><b>let </b><font color="Maroon" title="c5">la</font> be   <a href="ami_1.html#NM1" title="AMI_1:NM.1">Object</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Maroon" title="c5">la</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> implies for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font> holds  <font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Olive" title="b1">a</font> is   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <font color="Maroon" title="c3">S</font> )</span><br/>









<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E31">A1</font></i>: 
 <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <font color="Maroon" title="c5">la</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</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">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font> holds  <font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Olive" title="b1">a</font> is   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <font color="Maroon" title="c3">S</font></span><br/>

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">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="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">a</font> is   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <font color="Maroon" title="c3">S</font></span><br/>

<b>set </b><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">a</font>;<br/>
<a NAME="E2:64"/><i><font color="Green" title="E32">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">la</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E3:64"/><i><font color="Green" title="E33">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <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"> ( <font color="Maroon" title="c8">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"><font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )</span><br/><b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon" title="c8">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"><font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span><br/><a NAME="E2:64_1"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">la</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E2:64"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:64_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/><b>hence </b><a NAME="E4:64_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="ami_1.html#E1:64"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="ami_1.html#E2:64_1"><i><font color="Green" title="E34">A4</font></i></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>then reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K8" title="CARD_3:func.8">sproduct</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <font color="Maroon" title="c3">S</font> <b>by </b><i><a class="txt" href="ami_1.html#E2:64"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="ami_1.html#E3:64"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="card_3.html#D9" title="CARD_3:def.9">CARD_3:def 9</a></i>;<br/>
<a NAME="E6:64"/>
<font color="Maroon" title="c8">p</font> is   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <font color="Maroon" title="c3">S</font>
 
;<br/>
<b>hence </b><a NAME="E7:64"/>
<font color="Maroon" title="c5">la</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">a</font> is   <a href="ami_1.html#NM8" title="AMI_1:NM.8">FinPartState</a> of <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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