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<span class="kw">let </span><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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">m</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  &amp; <font color="Maroon" title="c2">S</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>  &amp; ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( ( <font color="Olive" title="b2">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">b</font> implies <font color="Olive" title="b1">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">b</font> ) &amp; ( <font color="Olive" title="b1">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">b</font> implies <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font> ) ) ) holds <br/><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">R</font>,<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"> ( <font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a>  &amp; <font color="Maroon" title="c2">S</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  &amp; <font color="Maroon" title="c2">S</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>  &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( ( <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> implies <font color="Maroon" title="c3">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font> ) &amp; ( <font color="Maroon" title="c3">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font> implies <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font> ) ) ) implies <font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="dickson.html#V3" title="DICKSON:attr.3">quasi_ordered</a> 
 <span class="kw">and </span><br/><a NAME="E2:32"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c2">S</font> is  <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> 
 <span class="kw">and </span><br/><a NAME="E3:32"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> 
 <span class="kw">and </span><br/><a NAME="E4:32"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( ( <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> implies <font color="Maroon" title="c3">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font> ) &amp; ( <font color="Maroon" title="c3">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font> implies <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</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="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">IR</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">IS</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font>;<br/>
<a NAME="E5:32"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="orders_2.html#D6" title="ORDERS_2:def.6">ORDERS_2:def 6</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> st <font color="Olive" title="b1">f</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</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="c6">f</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E3:32_1"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c6">f</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> 
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">f9</font> = <font color="Maroon" title="c6">f</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font> ;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <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> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> = <font color="Maroon" title="c3">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> $1</span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">g9</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E6:32_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
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>  holds  <font color="Maroon" title="c8">g9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g9</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <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="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">n1</font> = <font color="Maroon" title="c10">n</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E2:32_1_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></span>;<br/><a NAME="E3:32_1_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></span>;<br/><a NAME="E4:32_1_1"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
not <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E5:32_1_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E28">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_1_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font>
 ; <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="E2:32_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E5:32_1_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a></span>;<br/><a NAME="E3:32_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dickson.html#K1" title="DICKSON:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E20">A4</font></span></a></span>;<br/><a NAME="E4:32_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="dickson.html#D4" target="_self" title="DICKSON:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:32_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:32_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">fn1</font> = <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">fn</font> = <font color="Maroon" title="c6">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/><a NAME="E10:32_1_1"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c12">fn1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">fn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>;<br/><a NAME="E11:32_1_1"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<font color="Maroon" title="c8">g9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">fn1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a></span>;<br/><a NAME="E12:32_1_1"/>
<font color="Maroon" title="c8">g9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c11">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">fn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a></span>;<br/><a NAME="E13:32_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">g9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">g9</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E10:32_1_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="txt" href="#E11:32_1_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a></span>;<br/><a NAME="E14:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A15</font></span>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>;<br/><a NAME="E15:32_1_1"/><span class="kw">then </span>
not <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E6:32_1_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a></span>;<br/><a NAME="E16:32_1_1"/><span class="kw">then </span>
not <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E17:32_1_1"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32_1_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 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/></div><span class="kw">end;</span></div><a NAME="E9:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">g</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E10:32_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="wellfnd1.html#T15" title="WELLFND1:th.15">WELLFND1:15</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:32"/>
<font color="Maroon" title="c1">R</font> <a href="dickson.html#K5" title="DICKSON:func.5">\~</a>  is  <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellfnd1.html#T15" title="WELLFND1:th.15">WELLFND1:15</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/>


</div>
