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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="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">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>  st <font color="Olive" title="b2">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  holds <br/><font color="Olive" title="b1">A1</font>,<font color="Olive" title="b2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A1</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</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">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>  st <font color="Olive" title="b1">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  holds <br/><font color="Maroon" title="c3">A1</font>,<font color="Olive" title="b1">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A1</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <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>; <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">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  implies <font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A1</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:5"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</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>,<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A1</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> </span><br/>

<a NAME="E2:5"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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="#E1:5"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E5">A2</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c3">A1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<font color="Maroon" title="c3">A1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c3">A1</font></span>)</span> = 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K5" title="RELSET_1:func.5">|</a> <font color="Maroon" title="c3">A1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A1</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></span>;<br/>
<a NAME="E6:5"/>
<font color="Maroon" title="c5">g</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="#E1:5"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="metrizts.html#T2" target="_self" title="METRIZTS:th.2">Th2</a></span>;<br/>
<a NAME="E7:5"/><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>,<font color="Maroon" title="c2">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A1</font></span>)</span> <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="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:5"/>
<font color="Maroon" title="c3">A1</font>,<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A1</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metrizts.html#D1" target="_self" title="METRIZTS:def.1">Def1</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/>


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