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<span class="kw">let </span><font color="Maroon" title="c5">P1</font>, <font color="Maroon" title="c6">P2</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"> (  ex <font color="Olive" title="b1">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">KX</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">KX1</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">KX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Olive" title="b3">KX1</font>) ) ) &amp;  ex <font color="Olive" title="b1">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">KX</font> &amp; <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P2</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">KX1</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">KX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Olive" title="b3">KX1</font>) ) ) implies <font color="Maroon" title="c5">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c7">F1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E2:133_1_2"/><span class="lab"><font color="Green" title="E51">A39</font></span>: 
<font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">KX</font>
 <span class="kw">and </span><br/><a NAME="E3:133_1_2"/><span class="lab"><font color="Green" title="E52">A40</font></span>: 
<font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font>
 <span class="kw">and </span><br/><a NAME="E4:133_1_2"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E5:133_1_2"/><span class="lab"><font color="Green" title="E53">A41</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">KX1</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">KX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Olive" title="b2">KX1</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">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  holds <br/>(  not <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">KX</font> or  not <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P2</font> or  not  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  or  ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b3">KX1</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b3">KX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> &amp;  not <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Olive" title="b3">KX1</font>) ) ) or <font color="Maroon" title="c5">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c8">F2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><a NAME="E7:133_1_2"/><span class="lab"><font color="Green" title="E54">A42</font></span>: 
<font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">KX</font>
 <span class="kw">and </span><br/><a NAME="E8:133_1_2"/><span class="lab"><font color="Green" title="E55">A43</font></span>: 
<font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P2</font>
 <span class="kw">and </span><br/><a NAME="E9:133_1_2"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c8">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E10:133_1_2"/><span class="lab"><font color="Green" title="E56">A44</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">KX1</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">KX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Olive" title="b2">KX1</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" title="c5">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P2</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>( <font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp;  ex <font color="Olive" title="b1">KX1</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font> st <font color="Olive" title="b1">KX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E11:133_1_2"/><span class="lab"><font color="Green" title="E57">A45</font></span>: 
 for <font color="Olive" title="b1">k</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">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</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="133_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">k</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="c9">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:133_1_2_1"/><span class="lab"><font color="Green" title="E58">A46</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</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="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">KX1</font> being    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:133_1_2_1"/><span class="lab"><font color="Green" title="E59">A47</font></span>: 
<font color="Maroon" title="c10">KX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_1_2_1"><span class="lab"><font color="Green" title="E58">A46</font></span></a></span>;<br/>
<a NAME="E4:133_1_2_1"/>
<font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K11" title="SIMPLEX0:func.11">subdivision</a> (<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c10">KX1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:133_1_2"><span class="lab"><font color="Green" title="E53">A41</font></span></a>, <a class="txt" href="#E3:133_1_2_1"><span class="lab"><font color="Green" title="E59">A47</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:133_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</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="#E10:133_1_2"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="txt" href="#E1:133_1_2_1"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="txt" href="#E3:133_1_2_1"><span class="lab"><font color="Green" title="E59">A47</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>
<a NAME="E12:133_1_2"/><span class="lab"><font color="Green" title="E58">A48</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="txt" href="#E2:133_1_2"><span class="lab"><font color="Green" title="E51">A39</font></span></a>, <a class="txt" href="#E7:133_1_2"><span class="lab"><font color="Green" title="E54">A42</font></span></a></span>;<br/>
<a NAME="E13:133_1_2"/>
 for <font color="Olive" title="b1">k</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">k</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="#E12:133_1_2"><span class="lab"><font color="Green" title="E58">A48</font></span></a>, <a class="txt" href="#E11:133_1_2"><span class="lab"><font color="Green" title="E57">A45</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E14:133_1_2"/>
<font color="Maroon" title="c5">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:133_1_2"><span class="lab"><font color="Green" title="E52">A40</font></span></a>, <a class="txt" href="#E8:133_1_2"><span class="lab"><font color="Green" title="E55">A43</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>
