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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" 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"> ( <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>  iff  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font>, <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  )</span><br/>



<a NAME="E1:3"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
( <font color="Maroon" title="c1">T1</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font> #) &amp; <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"><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="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font> #) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T93" onmouseover="rs('tsep_1/T93')" onmouseout="rh()">TSEP_1:93</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:3_1"/>
(  not <font color="Maroon" title="c2">T2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <font color="Maroon" title="c2">T2</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="3_1_1"><span class="kw">suppose </span></a><a NAME="E1:3_1_1"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
 not <font color="Maroon" title="c2">T2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <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"> ( <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>  iff  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font>, <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:3_1_1"/>
( <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  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font>, <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E1:3_1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="topreala.html#T15" onmouseover="rs('topreala/T15')" onmouseout="rh()">TOPREALA:15</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"> (  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font>, <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  implies <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>  )</span><br/><span class="kw">assume </span><a NAME="E3:3_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font>, <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 ; <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"> <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> </span><br/><a NAME="E4:3_1_1"/><span class="kw">then </span>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font> #), <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</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="txt" href="#E1:3"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:3_1_1"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="topreala.html#T15" onmouseover="rs('topreala/T15')" onmouseout="rh()">TOPREALA:15</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2"><span class="kw">suppose </span></a><a NAME="E1:3_1_2"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
<font color="Maroon" title="c2">T2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <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"> ( <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>  iff  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font>, <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  )</span><br/><div class="add"><div><span class="kw">hereby </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"> (  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font>, <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a>  implies <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>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:3_1_2_1"/>
<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" 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">  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font>, <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> </span><br/><a NAME="E2:3_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">T1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  iff <font color="Maroon" title="c2">T2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow14.html#T18" onmouseover="rs('yellow14/T18')" onmouseout="rh()">YELLOW14:18</a></span>;<br/><a NAME="E3:3_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">T1</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <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>,<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"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</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="txt" href="#E1:3_1_2"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="txt" href="metrizts.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:3_1_2_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font>, <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 ; <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/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E3:3_1_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T1</font>, <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font> <a href="metrizts.html#R1" title="METRIZTS:pred.1">are_homeomorphic</a> 
 ; <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"> <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> </span><br/><a NAME="E4:3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">T1</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <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>,<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"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T2</font></span>)</span> <a href="borsuk_3.html#R1" title="BORSUK_3:pred.1">are_homeomorphic</a> 
 ;<br/><a NAME="E5:3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">T1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1_2"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="ref" href="yellow14.html#T18" onmouseover="rs('yellow14/T18')" onmouseout="rh()">YELLOW14:18</a></span>;<br/><span class="kw">hence </span><a NAME="E6:3_1_2"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1_2"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="txt" href="metrizts.html#E2"><span class="lab"><font color="Green" title="E1">Lm1</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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