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

<span class="kw">set </span><font color="Maroon" title="c4">iG1</font> = <span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">iG2</font> = <span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1;<br/>
<a NAME="E1:32_1_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#D6" target="_self" onmouseover="rs('topdim_1/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E2:32_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E3:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <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="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<a NAME="E4:32_1_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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="c3">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#D6" target="_self" onmouseover="rs('topdim_1/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E5:32_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E6:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <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="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<a NAME="E7:32_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E8:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:32_1_1"/>
( <span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:32_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E10:32_1_1"/><span class="kw">then </span>
 <a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c2">G1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topdim_1.html#K3" title="TOPDIM_1:func.3">ind</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:32_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">G2</font> holds <br/><font color="Olive">b<sub>1</sub></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="#E8:32_1_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="topdim_1.html#E22"><span class="lab"><font color="Green" title="E17">Lm4</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/>


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