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

<span class="kw">let </span><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font><br/> for <font color="Olive" title="b2">A2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font>  st <font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">A2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  holds <br/>( <font color="Olive" title="b1">A1</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  iff <font color="Olive" title="b2">A2</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>; <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">A2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font>  st <font color="Maroon" title="c3">A1</font>,<font color="Olive" title="b1">A2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  holds <br/>( <font color="Maroon" title="c3">A1</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  iff <font color="Olive" title="b1">A2</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</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="c3">A1</font>,<font color="Maroon" title="c4">A2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  implies ( <font color="Maroon" title="c3">A1</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  iff <font color="Maroon" title="c4">A2</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:51"/>
<font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c4">A2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 ; <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="c3">A1</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  iff <font color="Maroon" title="c4">A2</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  )</span><br/>

<a NAME="E2:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c1">T1</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c2">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A2</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#D1" title="METRIZTS:def.1">METRIZTS:def 1</a></span>;<br/>
<div><span class="kw">hereby </span> <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="c4">A2</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c3">A1</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:51_1"/>
<font color="Maroon" title="c3">A1</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> 
 ; <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="c4">A2</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> </span><br/><a NAME="E2:51_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">A2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="topdim_1.html#E38"><span class="lab"><font color="Green" title="E30">Lm9</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_1"/>
<font color="Maroon" title="c4">A2</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T18" target="_self" title="TOPDIM_1:th.18">Th18</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>

<span class="kw">assume </span><a NAME="E4:51"/>
<font color="Maroon" title="c4">A2</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> 
 ; <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="c3">A1</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> </span><br/>

<a NAME="E5:51"/><span class="kw">then </span>
<font color="Maroon" title="c1">T1</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">A1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="topdim_1.html#E38"><span class="lab"><font color="Green" title="E30">Lm9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51"/>
<font color="Maroon" title="c3">A1</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T18" target="_self" title="TOPDIM_1:th.18">Th18</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>
