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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">i1</font>, <font color="Olive" title="b2">i2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/> for <font color="Olive" title="b3">KX</font> being   <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i1</font> &amp; <font color="Olive" title="b1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i2</font> holds <br/> <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Olive" title="b3">KX</font>,<font color="Olive" title="b1">i1</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Olive" title="b3">KX</font>,<font color="Olive" title="b2">i2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">i1</font>, <font color="Maroon" title="c3">i2</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">KX</font> being   <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>  st  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font> &amp; <font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font> holds <br/> <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Olive" title="b1">KX</font>,<font color="Maroon" title="c2">i1</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Olive" title="b1">KX</font>,<font color="Maroon" title="c3">i2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font> &amp; <font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font> implies  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c4">KX</font>,<font color="Maroon" title="c2">i1</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c4">KX</font>,<font color="Maroon" title="c3">i2</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:109"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font>
 <span class="kw">and </span><br/><a NAME="E2:109"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c2">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c4">KX</font>,<font color="Maroon" title="c2">i1</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c4">KX</font>,<font color="Maroon" title="c3">i2</font></span><br/>

<a NAME="E3:109"/>
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:109"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E4:109"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E5:109"/>
<span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:109"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">I1</font> = <font color="Maroon" title="c2">i1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1, <font color="Maroon" title="c6">I2</font> = <font color="Maroon" title="c3">i2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 as    <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">by</span> <span class="lab"><a class="txt" href="#E4:109"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/>
<a NAME="E7:109"/>
 <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">KX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">KX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">KX</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">KX</font> )</span><br/>

<a NAME="E1:109_1"/>
<font color="Maroon" title="c5">I1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">I2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E2:109_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A4</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">I1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">I2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T41" title="NAT_1:th.41">NAT_1:41</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:109_1"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">KX</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" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">KX</font></span><br/>

<a NAME="E4:109_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">I1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D2" target="_self" title="SIMPLEX0:def.2">Def2</a></span>;<br/>
<a NAME="E5:109_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A6</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">I2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:109_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E6:109_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">KX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:109_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="simplex0.html#D2" target="_self" title="SIMPLEX0:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:109_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">KX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:109_1"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="simplex0.html#D2" target="_self" title="SIMPLEX0:def.2">Def2</a></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><span class="kw">end;</span></div>
<a NAME="E8:109"/><span class="kw">then </span>
 <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">KX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a>  <a href="simplex0.html#NK4" title="SIMPLEX0:NK.4">the_subsets_with_limited_card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i2</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">KX</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T17" title="SETFAM_1:th.17">SETFAM_1:17</a></span>;<br/>
<a NAME="E9:109"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A7</font></span>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c4">KX</font>,<font color="Maroon" title="c2">i1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c4">KX</font>,<font color="Maroon" title="c3">i2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T6" target="_self" title="SIMPLEX0:th.6">Th6</a></span>;<br/>
<a NAME="E10:109"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c4">KX</font>,<font color="Maroon" title="c2">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c4">KX</font>,<font color="Maroon" title="c3">i2</font></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E11:109"/>
 <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c4">KX</font>,<font color="Maroon" title="c2">i1</font> is    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of  <a href="simplex0.html#K9" title="SIMPLEX0:func.9">Skeleton_of</a> <font color="Maroon" title="c4">KX</font>,<font color="Maroon" title="c3">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:109"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="simplex0.html#D13" target="_self" title="SIMPLEX0:def.13">Def13</a></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>
