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<span class="kw">let </span><font color="Maroon" title="c1">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>  st  <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">P1</font> =  <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> 1,<font color="Maroon" title="c1">m</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="heyting3.html#T28" target="_self" title="HEYTING3:th.28">Th28</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c3">a</font> implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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><br/>

<span class="kw">assume </span><a NAME="E2:63"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c3">a</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">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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><br/>

<a NAME="E3:63"/>
 <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> 1,<font color="Maroon" title="c1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="heyting3.html#D5" target="_self" title="HEYTING3:def.5">Def5</a></span>;<br/>
<a NAME="E4:63"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:63"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:63"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c3">a</font> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:63"/>
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></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="heyting3.html#T39" target="_self" title="HEYTING3:th.39">Th39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:63"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:63"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="heyting3.html#T30" target="_self" title="HEYTING3:th.30">Th30</a>, <a class="ref" href="waybel15.html#T5" title="WAYBEL15:th.5">WAYBEL15:5</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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