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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">KX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">SX</font> being    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Olive" title="b2">KX</font><br/>  for <font color="Olive" title="b4">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Olive" title="b1">n</font>,<font color="Olive" title="b4">P</font>,<font color="Olive" title="b3">SX</font>) is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Olive" title="b1">n</font>,<font color="Olive" title="b4">P</font>,<font color="Olive" title="b2">KX</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">KX</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">SX</font> being    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Olive" title="b1">KX</font><br/>  for <font color="Olive" title="b3">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c2">n</font>,<font color="Olive" title="b3">P</font>,<font color="Olive" title="b2">SX</font>) is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c2">n</font>,<font color="Olive" title="b3">P</font>,<font color="Olive" title="b1">KX</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">KX</font> be    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</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">SX</font> being    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c3">KX</font><br/>  for <font color="Olive" title="b2">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c2">n</font>,<font color="Olive" title="b2">P</font>,<font color="Olive" title="b1">SX</font>) is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c2">n</font>,<font color="Olive" title="b2">P</font>,<font color="Maroon" title="c3">KX</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">SX</font> be    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c3">KX</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">P</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds   <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c4">SX</font>) is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c3">KX</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">P</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c4">SX</font>) is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">KX</font>)</span><br/>









<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> ($1,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c4">SX</font>) is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> ($1,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">KX</font>);<br/>
<a NAME="E1:138"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="138_1"><span class="kw">proof </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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:138_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">Pn</font> =  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c4">SX</font>) as    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">KX</font>) ;<br/>
<a NAME="E3:138_1"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E57">A3</font></span>:  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</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="c5">P</font>,<font color="Maroon" title="c4">SX</font>) = 
 <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (1,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c7">Pn</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T63" target="_self" onmouseover="rs('simplex0/T63')" onmouseout="rh()">Th63</a></span>
<br/>.= 
 <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c7">Pn</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T62" target="_self" onmouseover="rs('simplex0/T62')" onmouseout="rh()">Th62</a></span>
;<br/>
 <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</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="c5">P</font>,<font color="Maroon" title="c3">KX</font>) = 
 <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (1,<font color="Maroon" title="c5">P</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">KX</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:138_1"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="simplex0.html#T63" target="_self" onmouseover="rs('simplex0/T63')" onmouseout="rh()">Th63</a></span>
<br/>.= 
 <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c5">P</font>,<span class="p1">(<span class="default"><a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">KX</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T62" target="_self" onmouseover="rs('simplex0/T62')" onmouseout="rh()">Th62</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:138_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:138_1"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="simplex0.html#T58" target="_self" onmouseover="rs('simplex0/T58')" onmouseout="rh()">Th58</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>
<a NAME="E2:138"/>
 <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c4">SX</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">SX</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T61" target="_self" onmouseover="rs('simplex0/T61')" onmouseout="rh()">Th61</a></span>;<br/>
<a NAME="E3:138"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T61" target="_self" onmouseover="rs('simplex0/T61')" onmouseout="rh()">Th61</a></span>;<br/>
<a NAME="E4:138"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E3:138"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="txt" href="#E1:138"><span class="lab"><font color="Green" title="E55">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:138"/>
 <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c4">SX</font>) is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K12" title="SIMPLEX0:func.12">subdivision</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c3">KX</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"> verum</span><br/>


</div>
