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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>,<font color="Olive" title="b2">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="Olive" title="b1">p</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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>,<font color="Olive" title="b1">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="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</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>,<font color="Maroon" title="c3">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="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) )</span><br/>





<a NAME="E1:7"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 <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="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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>)</span> <a href="fin_topo.html#K12" title="FIN_TOPO:func.12">^f</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo3.html#T48" title="FINTOPO3:th.48">FINTOPO3:48</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:7"/>
<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>,<font color="Maroon" title="c3">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="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span><br/>

<span class="kw">hence </span><a NAME="E3:7"/>
 <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 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="c3">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="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="fintopo3.html#T1" title="FINTOPO3:th.1">FINTOPO3:1</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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