<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">G</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( (  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Olive" title="b1">G</font> is  <a href="topdim_1.html#V2" title="TOPDIM_1:attr.2">with_finite_small_inductive_dimension</a>  ) iff <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</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"> ( (  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c2">G</font> is  <a href="topdim_1.html#V2" title="TOPDIM_1:attr.2">with_finite_small_inductive_dimension</a>  ) iff <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>



<a NAME="E1:29"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="prob_1.html#K13" title="PROB_1:func.13">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#D1" target="_self" title="TOPDIM_1:def.1">Def1</a></span>;<br/>
<a NAME="E2:29"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
;<br/>
<span class="kw">hence </span><a NAME="E3:29"/>
(  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c2">G</font> is  <a href="topdim_1.html#V2" title="TOPDIM_1:attr.2">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="topdim_1.html#D6" target="_self" title="TOPDIM_1:def.6">Def6</a></span>; <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="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies (  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c2">G</font> is  <a href="topdim_1.html#V2" title="TOPDIM_1:attr.2">with_finite_small_inductive_dimension</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:29"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c2">G</font> is  <a href="topdim_1.html#V2" title="TOPDIM_1:attr.2">with_finite_small_inductive_dimension</a>  )</span><br/>

<a NAME="E5:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c2">G</font> is  <a href="topdim_1.html#V2" title="TOPDIM_1:attr.2">with_finite_small_inductive_dimension</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="topdim_1.html#D3" target="_self" title="TOPDIM_1:def.3">Def3</a></span>;<br/>
<a NAME="E6:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#D6" target="_self" title="TOPDIM_1:def.6">Def6</a></span>;<br/>
<a NAME="E7:29"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
;<br/>
<a NAME="E8:29"/><span class="kw">then </span>
 <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="topdim_1.html#D6" target="_self" title="TOPDIM_1:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:29"/>
(  <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c2">G</font> is  <a href="topdim_1.html#V2" title="TOPDIM_1:attr.2">with_finite_small_inductive_dimension</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E6:29"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="topdim_1.html#D3" target="_self" title="TOPDIM_1:def.3">Def3</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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>
