<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">R</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font><br/>  for <font color="Olive" title="b2">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">s</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>  holds <br/><font color="Olive" title="b1">s</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Olive" title="b2">j</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">s1</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c2">s1</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>  holds <br/><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Olive" title="b1">j</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">s1</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>  implies <font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:69"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c2">s1</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">s2</font> = <font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">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/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI: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> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI: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> )</span><br/><span class="kw">set </span><font color="Maroon" title="c7">nj</font> = <font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">j</font>;<br/><a NAME="E1:69_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" onmouseover="rs('nat_1/D3')" onmouseout="rh()">NAT_1:def 3</a></span>;<br/><span class="lab"><font color="Green" title="E42">A3</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D3" onmouseover="rs('nat_1/D3')" onmouseout="rh()">NAT_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

;<br/><span class="kw">hence </span><a NAME="E3:69_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI: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></span><br/><span class="kw">thus </span><a NAME="E4:69_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">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"><span class="p3">(<span class="default"><font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">n</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E2:69_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:69"/>
<font color="Maroon" title="c2">s1</font> <a href="nat_1.html#K7" title="NAT_1:func.7">^\</a> <font color="Maroon" title="c3">j</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
