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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b2">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" title="c1">T</font> is  <a href="fin_topo.html#NV1" title="FIN_TOPO:NV.1">filled</a>  holds <br/> <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Olive" title="b1">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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" title="c1">T</font> is  <a href="fin_topo.html#NV1" title="FIN_TOPO:NV.1">filled</a>  holds <br/> <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">k</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is  <a href="fin_topo.html#NV1" title="FIN_TOPO:NV.1">filled</a>  implies  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">k</font>) )</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> <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,$1);<br/>
<span class="kw">assume </span><a NAME="E1:8"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="fin_topo.html#NV1" title="FIN_TOPO:NV.1">filled</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<a NAME="E2:8"/><span class="lab"><font color="Green" title="E8">A2</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="8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:8_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:8_1"/>
 <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">k</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="fintopo5.html#T7" target="_self" title="FINTOPO5:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:8_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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/>


</div><span class="kw">end;</span></div>
<a NAME="E3:8"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E4:8"/>
 for <font color="Olive" title="b1">i</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">i</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="#E3:8"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E8">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:8"/>
 <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo3.html#K10" title="FINTOPO3:func.10">U_FT</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">k</font>)
 ; <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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