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<span class="kw">let </span><font color="Maroon" title="c5">it1</font>, <font color="Maroon" title="c6">it2</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="extpro_1.html#K3" title="EXTPRO_1:func.3">CurInstr</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c5">it1</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c2">S</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <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>   st  <a href="extpro_1.html#K3" title="EXTPRO_1:func.3">CurInstr</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Olive" title="b1">k</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c5">it1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) &amp;  <a href="extpro_1.html#K3" title="EXTPRO_1:func.3">CurInstr</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">it2</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c2">S</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <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>   st  <a href="extpro_1.html#K3" title="EXTPRO_1:func.3">CurInstr</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Olive" title="b1">k</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c6">it2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) implies <font color="Maroon" title="c5">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">it2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_1_2"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
(  <a href="extpro_1.html#K3" title="EXTPRO_1:func.3">CurInstr</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c5">it1</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c2">S</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <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>   st  <a href="extpro_1.html#K3" title="EXTPRO_1:func.3">CurInstr</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Olive" title="b1">k</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c5">it1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) &amp;  <a href="extpro_1.html#K3" title="EXTPRO_1:func.3">CurInstr</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">it2</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c2">S</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <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>   st  <a href="extpro_1.html#K3" title="EXTPRO_1:func.3">CurInstr</a> (<font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">s</font>,<font color="Olive" title="b1">k</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="compos_1.html#K2" title="COMPOS_1:func.2">halt</a> <font color="Maroon" title="c2">S</font> holds <br/><font color="Maroon" title="c6">it2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) &amp;  not <font color="Maroon" title="c5">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">it2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:50_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">it1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">it2</font> &amp; <font color="Maroon" title="c6">it2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">it1</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:50_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_1_2"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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