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<b>let </b><font color="Maroon" title="c5">IT1</font>, <font color="Maroon" title="c6">IT2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<a NAME="E1:7_1_2"/><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Q</font> &amp; <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>assume </b><a NAME="E2:7_1_2"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c5">IT1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> &amp; ( for <font color="Olive" title="b1">g1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span><br/> for <font color="Olive" title="b2">s1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">g1</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Olive" title="b1">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Olive" title="b1">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT1</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s1</font> &amp; <font color="Olive" title="b2">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/>for <font color="Olive" title="b3">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">t</font> &amp; <font color="Olive" title="b3">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">s1</font> holds <br/> not <font color="Olive" title="b1">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font> ) )
 ;<br/>

<b>assume </b><a NAME="E3:7_1_2"/><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c6">IT2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</font> &amp; ( for <font color="Olive" title="b1">g2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span><br/> for <font color="Olive" title="b2">s2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">g2</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Olive" title="b1">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Olive" title="b1">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Olive" title="b1">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s2</font> &amp; <font color="Olive" title="b2">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/>for <font color="Olive" title="b3">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">t</font> &amp; <font color="Olive" title="b3">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">s2</font> holds <br/> not <font color="Olive" title="b1">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font> ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span><b> such that </b><br/><a NAME="E5:7_1_2"/><i><font color="Green" title="E10">A6</font></i>: 
( <font color="Maroon" title="c7">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="txt" href="jordan5c.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<i><font color="Green" title="E11">A7</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">g</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan5c.html#E5:7_1_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<i><font color="Green" title="E12">A8</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">g</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
<b>by </b><i><a class="txt" href="jordan5c.html#E5:7_1_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 

;<br/>
<b>consider </b><font color="Maroon" title="c8">ss1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:7_1_2"/><i><font color="Green" title="E13">A9</font></i>: 
( <font color="Maroon" title="c8">ss1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">g</font> &amp; <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">ss1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT1</font> )
 <b>by </b><i><a class="txt" href="jordan5c.html#E1:7_1_2"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="jordan5c.html#E2:7_1_2"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="jordan5c.html#E6:7_1_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">ss1</font> = <font color="Maroon" title="c8">ss1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan5c.html#E7:7_1_2"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="jordan5c.html#E9:7_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">ss2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:7_1_2"/><i><font color="Green" title="E14">A10</font></i>: 
( <font color="Maroon" title="c10">ss2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">g</font> &amp; <font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">ss2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font> )
 <b>by </b><i><a class="txt" href="jordan5c.html#E1:7_1_2"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="jordan5c.html#E3:7_1_2"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="jordan5c.html#E6:7_1_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">ss2</font> = <font color="Maroon" title="c10">ss2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan5c.html#E7:7_1_2"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="jordan5c.html#E12:7_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/>
<a NAME="E14:7_1_2"/><i><font color="Green" title="E15">A11</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">ss1</font> &amp; <font color="Maroon" title="c9">ss1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">ss2</font> &amp; <font color="Maroon" title="c11">ss2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan5c.html#E9:7_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan5c.html#E12:7_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_2_3"/>
( <font color="Maroon" title="c9">ss1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">ss2</font> or <font color="Maroon" title="c9">ss1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">ss2</font> or <font color="Maroon" title="c9">ss1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">ss2</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_3_1"><b>suppose </b></a><a NAME="E1:7_1_2_3_1"/>
<font color="Maroon" title="c9">ss1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">ss2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_2_3_1"/>
<font color="Maroon" title="c5">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font>
 <b>by </b><i><a class="txt" href="jordan5c.html#E1:7_1_2"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="jordan5c.html#E2:7_1_2"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="jordan5c.html#E3:7_1_2"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="jordan5c.html#E5:7_1_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="jordan5c.html#E9:7_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan5c.html#E12:7_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan5c.html#E14:7_1_2"><i><font color="Green" title="E15">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_3_2"><b>suppose </b></a><a NAME="E1:7_1_2_3_2"/>
<font color="Maroon" title="c9">ss1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">ss2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_2_3_2"/>
<font color="Maroon" title="c5">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font>
 <b>by </b><i><a class="txt" href="jordan5c.html#E9:7_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan5c.html#E12:7_1_2"><i><font color="Green" title="E14">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_3_3"><b>suppose </b></a><a NAME="E1:7_1_2_3_3"/>
<font color="Maroon" title="c9">ss1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">ss2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_2_3_3"/>
<font color="Maroon" title="c5">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font>
 <b>by </b><i><a class="txt" href="jordan5c.html#E1:7_1_2"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="jordan5c.html#E2:7_1_2"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="jordan5c.html#E3:7_1_2"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="jordan5c.html#E5:7_1_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="jordan5c.html#E9:7_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan5c.html#E12:7_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan5c.html#E14:7_1_2"><i><font color="Green" title="E15">A11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
