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<span class="kw">let </span><font color="Maroon" title="c1">T</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">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">n</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   <a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span>)</span>,<font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">n</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</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">n</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   <a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span>,<font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Maroon" title="c3">B</font>,<font color="Olive" title="b1">n</font></span>)</span></span><br/>





<span class="kw">defpred </span><font color="Maroon">S<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">means </span><span class="p1">(<span class="default"><a href="fintopo3.html#K8" title="FINTOPO3:func.8">Fdfl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" title="FINTOPO3:func.8">Fdfl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" title="FINTOPO3:func.8">Fdfl</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">n</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">  <a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">n</font></span>)</span></span><br/>

<a NAME="E1:43"/><span class="lab"><font color="Green" title="E24">A1</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="p1">(<span class="default"><a href="fintopo3.html#K8" title="FINTOPO3:func.8">Fdfl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="fintopo3.html#K1" title="FINTOPO3:func.1">^d</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo3.html#D8" target="_self" title="FINTOPO3:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="fintopo3.html#K1" title="FINTOPO3:func.1">^d</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="fintopo3.html#K1" title="FINTOPO3:func.1">^d</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="fintopo3.html#T12" target="_self" title="FINTOPO3:th.12">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Maroon" title="c2">A</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c5">k</font></span>)</span> <a href="fintopo3.html#K1" title="FINTOPO3:func.1">^d</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo3.html#D8" target="_self" title="FINTOPO3:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" title="FINTOPO3:func.8">Fdfl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" title="FINTOPO3:func.8">Fdfl</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo3.html#D8" target="_self" title="FINTOPO3:def.8">Def8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:43_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <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="p1">(<span class="default"><a href="fintopo3.html#K8" title="FINTOPO3:func.8">Fdfl</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo3.html#D8" target="_self" title="FINTOPO3:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" title="FINTOPO3:func.8">Fdfl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo3.html#D8" target="_self" title="FINTOPO3:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" title="FINTOPO3:func.8">Fdfl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fintopo3.html#K8" title="FINTOPO3:func.8">Fdfl</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo3.html#D8" target="_self" title="FINTOPO3:def.8">Def8</a></span>
;<br/>
<a NAME="E3:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E4:43"/>
for <font color="Olive" title="b1">n</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">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E24">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:43"/>
 <a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c4">n</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K9" title="FINTOPO3:func.9">Fdfl</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">n</font></span>)</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>
