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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#V5" title="AMI_1:attr.5">IC-Ins-separated</a>   <a href="ami_1.html#V8" title="AMI_1:attr.8">definite</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">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Olive" title="b1">S</font>  st not  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; ( for <font color="Olive" title="b3">l</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>  holds   <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">l</font> <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="Olive" title="b3">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) holds <br/> <a href="amistd_1.html#K2" title="AMISTD_1:func.2">JUMP</a> <font color="Olive" title="b2">i</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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#V5" title="AMI_1:attr.5">IC-Ins-separated</a>   <a href="ami_1.html#V8" title="AMI_1:attr.8">definite</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">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <font color="Maroon" title="c2">S</font>  st not  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; ( for <font color="Olive" title="b2">l</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>  holds   <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">l</font> <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="Olive" title="b2">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) holds <br/> <a href="amistd_1.html#K2" title="AMISTD_1:func.2">JUMP</a> <font color="Olive" title="b1">i</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</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"> ( not  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; ( for <font color="Olive" title="b1">l</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>  holds   <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">l</font> <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="Olive" title="b1">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) implies  <a href="amistd_1.html#K2" title="AMISTD_1:func.2">JUMP</a> <font color="Maroon" title="c3">i</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>





<span class="kw">given </span><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</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> <span class="kw"> such that </span><a NAME="E2:25"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_8.html#D1" title="YELLOW_8:def.1">YELLOW_8:def 1</a> <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">l</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 not  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">l</font> <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="Olive" title="b1">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  <a href="amistd_1.html#K2" title="AMISTD_1:func.2">JUMP</a> <font color="Maroon" title="c3">i</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <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>  : verum </span>}</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c5">q</font> as    <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>  ;<br/>
<span class="kw">assume </span><a NAME="E4:25"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
for <font color="Olive" title="b1">l</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>  holds   <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">l</font> <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="Olive" title="b1">l</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="amistd_1.html#K2" title="AMISTD_1:func.2">JUMP</a> <font color="Maroon" title="c3">i</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<span class="kw">assume </span><a NAME="E5:25"/>
not  <a href="amistd_1.html#K2" title="AMISTD_1:func.2">JUMP</a> <font color="Maroon" title="c3">i</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:25"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <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>  : verum </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E8:25"/>
 <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c7">p</font> <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="c7">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>;<br/>
<a NAME="E9:25"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <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>  : verum </span>}</span> 
 
;<br/>
<a NAME="E10:25"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<a NAME="E11:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font>
 
<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/>
<a NAME="E12:25"/>
 <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c8">q</font> <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="c8">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>;<br/>
<a NAME="E13:25"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">l</font></span>)</span> where <font color="Olive" title="b1">l</font> is    <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>  : verum </span>}</span> 
 
;<br/>
<a NAME="E14:25"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:25"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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/>


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