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<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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a>  implies ( ( <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  ) &amp; ( <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  ) &amp; ( <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:49"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 ; <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="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  ) &amp; ( <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  ) &amp; ( <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font><span class="kw"> such that </span><br/><a NAME="E3:49"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a></span>;<br/>
<a NAME="E4:49"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E5:49"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:49_1"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T1</font> or  not <font color="Maroon" title="c1">T1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><span class="kw">suppose </span></a><a NAME="E1:49_1_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T1</font>
 ; <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="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  ) &amp; ( <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  ) &amp; ( <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font> ) )</span><br/><div class="add"><a NAME="E2:49_1_1"/><span class="kw">then </span>
 <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E5:49"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="topdim_1.html#T6" target="_self" title="TOPDIM_1:th.6">Th6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:49_1_1"/>
( ( <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  ) &amp; ( <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  ) &amp; ( <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E1:49_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="topdim_1.html#D4" target="_self" title="TOPDIM_1:def.4">Def4</a>, <a class="ref" href="topdim_1.html#T6" target="_self" title="TOPDIM_1:th.6">Th6</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><span class="kw">suppose </span></a><a NAME="E1:49_1_2"/>
 not <font color="Maroon" title="c1">T1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <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="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  ) &amp; ( <font color="Maroon" title="c2">T2</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  ) &amp; ( <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">t1</font> = <font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c5">t2</font> = <font color="Maroon" title="c2">T2</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E33">A3</font></span></a></span>;<br/><a NAME="E3:49_1_2"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c5">t2</font>,<font color="Maroon" title="c4">t1</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:49_1_2"/>
( <font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  iff <font color="Maroon" title="c2">T2</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="topdim_1.html#E37"><span class="lab"><font color="Green" title="E29">Lm8</font></span></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="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  implies  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font> )</span><br/><span class="kw">assume </span><a NAME="E5:49_1_2"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<font color="Maroon" title="c1">T1</font> is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a> 
 ; <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#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font></span><br/><a NAME="E6:49_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">T2</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="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="topdim_1.html#E37"><span class="lab"><font color="Green" title="E29">Lm8</font></span></a></span>;<br/><a NAME="E7:49_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</font></span>: 
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49_1_2"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="topdim_1.html#E37"><span class="lab"><font color="Green" title="E29">Lm8</font></span></a></span>;<br/><a NAME="E8:49_1_2"/>
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E5:49_1_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="topdim_1.html#E37"><span class="lab"><font color="Green" title="E29">Lm8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:49_1_2"/>
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c2">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <font color="Maroon" title="c1">T1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_1_2"><span class="lab"><font color="Green" title="E37">A8</font></span></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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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