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<b>let </b><font color="Maroon" title="c8">Y</font> be  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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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 <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span><br/> for <font color="Olive" title="b2">Ft</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b2">Ft</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">Ft</font> holds <br/> ex <font color="Olive" title="b3">G</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Olive" title="b3">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">G</font> &amp; <font color="Olive" title="b3">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Ft</font> &amp; ( for <font color="Olive" title="b4">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b4">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">H</font> &amp; <font color="Olive" title="b4">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Ft</font> holds <br/><font color="Olive" title="b3">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">H</font> ) )</span><br/><b>let </b><font color="Maroon" title="c9">F</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Ft</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c9">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">Ft</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">Ft</font> holds <br/> ex <font color="Olive" title="b2">G</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Olive" title="b2">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">G</font> &amp; <font color="Olive" title="b2">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Ft</font> &amp; ( for <font color="Olive" title="b3">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b3">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b3">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Ft</font> holds <br/><font color="Olive" title="b2">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">H</font> ) )</span><br/><b>let </b><font color="Maroon" title="c10">Ft</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c10">Ft</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">Ft</font> implies  ex <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b1">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> &amp; ( for <font color="Olive" title="b2">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b2">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">H</font> &amp; <font color="Olive" title="b2">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> holds <br/><font color="Olive" title="b1">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">H</font> ) ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c9">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c10">Ft</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">Ft</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b1">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> &amp; ( for <font color="Olive" title="b2">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b2">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">H</font> &amp; <font color="Olive" title="b2">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> holds <br/><font color="Olive" title="b1">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">H</font> ) )</span><br/>

<a NAME="E4:37"/><i><font color="Green" title="E40">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E5:37"/><i><font color="Green" title="E41">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E6:37"/><b>then </b><i><font color="Green" title="E42">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">Ft</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">UL</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E8:37"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c11">UL</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E9:37"/><i><font color="Green" title="E44">A8</font></i>: 
<font color="Maroon" title="c11">UL</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>and </b><br/><a NAME="E10:37"/><i><font color="Green" title="E45">A9</font></i>: 
for <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>  st <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">UL</font> holds <br/> ex <font color="Olive" title="b2">D</font> being <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">U</font> &amp; ( for <font color="Olive" title="b3">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">U</font></span>)</span>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">d</font> holds <br/><font color="Olive" title="b4">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> ) )
 <b>by </b><i><a class="txt" href="topalg_5.html#E38"><i><font color="Green" title="E32">Lm13</font></i></a></i>;<br/>
<i><font color="Green" title="E46">A10</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">F</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c8">Y</font> ex <font color="Olive" title="b2">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ex <font color="Olive" title="b3">N</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> ex <font color="Olive" title="b4">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">N</font> &amp; <font color="Olive" title="b4">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b3">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">Fn</font> &amp; <font color="Olive" title="b4">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b3">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; ( for <font color="Olive" title="b5">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b5">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b3">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">H</font> &amp; <font color="Olive" title="b5">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b3">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Olive" title="b4">Fn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b5">H</font> ) );<br/>
<a NAME="E12:37"/><i><font color="Green" title="E47">A11</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>  ex <font color="Olive" title="b2">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<b>consider </b><font color="Maroon" title="c13">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c14">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E2:37_2"/><i><font color="Green" title="E48">A12</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">TT</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:37_2"/><i><font color="Green" title="E49">A13</font></i>: 
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E5:37_2"/><i><font color="Green" title="E50">A14</font></i>: 
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">TT</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E6:37_2"/><i><font color="Green" title="E51">A15</font></i>: 
<font color="Maroon" title="c15">TT</font> is <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>
 <b>and </b><br/><a NAME="E7:37_2"/><i><font color="Green" title="E52">A16</font></i>: 
 ex <font color="Olive" title="b1">N</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> st <br/>( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; ( for <font color="Olive" title="b2">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> &amp; <font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> holds <br/> ex <font color="Olive" title="b3">Ui</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive" title="b3">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">UL</font> &amp; <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Ui</font> ) ) )
 <b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="#E8:37"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E9:37"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="topalg_5.html#T21" target="_self" title="TOPALG_5:th.21">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">N</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font><b> such that </b><br/><a NAME="E9:37_2"/><i><font color="Green" title="E53">A17</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N</font>
 <b>and </b><br/><a NAME="E10:37_2"/><i><font color="Green" title="E54">A18</font></i>: 
for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> holds <br/> ex <font color="Olive" title="b2">Ui</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive" title="b2">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">UL</font> &amp; <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Ui</font> )
 <b>by </b><i><a class="txt" href="#E7:37_2"><i><font color="Green" title="E52">A16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">N</font> = <font color="Maroon" title="c16">N</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> <b>by </b><i><a class="txt" href="#E9:37_2"><i><font color="Green" title="E53">A17</font></i></a></i>;<br/>
<a NAME="E12:37_2"/><i><font color="Green" title="E55">A19</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E13:37_2"/><i><font color="Green" title="E56">A20</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">TT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<div><i><font color="Green" title="E57">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; ( <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> implies ( <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) ) ) )</span><br/><b>assume </b><a NAME="E1:37_2_1"/><i><font color="Green" title="E58">A22</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; ( <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> implies ( <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) ) )</span><br/><a NAME="E2:37_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="#E1:37_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:37_2_1"/><b>then </b>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">i</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E4:37_2_1"/><i><font color="Green" title="E59">A23</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="#E4:37_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="#E6:37_2"><i><font color="Green" title="E51">A15</font></i></a>, <a class="txt" href="#E12:37_2"><i><font color="Green" title="E55">A19</font></i></a>, <a class="txt" href="#E1:37_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> implies ( <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )</span><br/><b>assume </b><a NAME="E5:37_2_1"/><i><font color="Green" title="E60">A24</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><a NAME="E6:37_2_1"/><i><font color="Green" title="E61">A25</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E7:37_2_1"/><i><font color="Green" title="E62">A26</font></i>: 
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E6:37_2"><i><font color="Green" title="E51">A15</font></i></a>, <a class="txt" href="#E1:37_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="txt" href="#E5:37_2_1"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><a NAME="E8:37_2_1"/>
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">TT</font>
 <b>by </b><i><a class="txt" href="#E5:37_2_1"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:37_2_1"/><b>then </b>
( <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">TT</font> or <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">TT</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E10:37_2_1"/>
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="#E5:37_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="#E6:37_2"><i><font color="Green" title="E51">A15</font></i></a>, <a class="txt" href="#E13:37_2"><i><font color="Green" title="E56">A20</font></i></a>, <a class="txt" href="#E5:37_2_1"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><b>hence </b><a NAME="E11:37_2_1"/>
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="#E7:37_2_1"><i><font color="Green" title="E62">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><b>thus </b><a NAME="E12:37_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E6:37_2"><i><font color="Green" title="E51">A15</font></i></a>, <a class="txt" href="#E1:37_2_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="txt" href="#E4:37_2_1"><i><font color="Green" title="E59">A23</font></i></a>, <a class="txt" href="#E5:37_2_1"><i><font color="Green" title="E60">A24</font></i></a>, <a class="txt" href="#E6:37_2_1"><i><font color="Green" title="E61">A25</font></i></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E58">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 implies <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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>  )</span><br/><b>assume </b><a NAME="E1:37_2_2"/><i><font color="Green" title="E59">A28</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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> </span><br/><b>thus </b><a NAME="E2:37_2_2"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> or <font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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>  )</span><br/>

<b>assume </b><a NAME="E1:37_2_2_1"/><i><font color="Green" title="E60">A29</font></i>: 
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> </span><br/>

<b>then reconsider </b><font color="Maroon" title="c20">a</font> = <font color="Maroon" title="c19">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E3:37_2_2_1"/>
( <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">a</font> &amp; <font color="Maroon" title="c20">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:37_2_2_1"><i><font color="Green" title="E60">A29</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E4:37_2_2_1"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">a</font> &amp; <font color="Maroon" title="c20">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E1:37_2_2"><i><font color="Green" title="E59">A28</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:37_2_2_1"/>
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> implies  ex <font color="Olive" title="b1">N2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> ex <font color="Olive" title="b2">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ex <font color="Olive" title="b3">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N2</font> &amp; <font color="Olive" title="b1">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Olive" title="b3">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">N2</font>,<font color="Olive" title="b2">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">Fn</font> &amp; <font color="Olive" title="b3">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) );<br/>
<a NAME="E16:37_2"/><i><font color="Green" title="E59">A30</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26" title="FINSEQ_3:th.26">FINSEQ_3:26</a></i>;<br/>
<a NAME="E17:37_2"/><i><font color="Green" title="E60">A31</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">i</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37_2_3"/><i><font color="Green" title="E61">A32</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">i</font>]
 <b>and </b><br/><a NAME="E2:37_2_3"/><i><font color="Green" title="E62">A33</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">N2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> ex <font color="Olive" title="b2">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ex <font color="Olive" title="b3">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N2</font> &amp; <font color="Olive" title="b1">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Olive" title="b3">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">N2</font>,<font color="Olive" title="b2">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">Fn</font> &amp; <font color="Olive" title="b3">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_3_1"/>
( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> )
 </span><b>by </b><i><a class="txt" href="#E2:37_2_3"><i><font color="Green" title="E62">A33</font></i></a>, <a class="ref" href="topreala.html#T23" title="TOPREALA:th.23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_3_1_1"><b>suppose </b></a><a NAME="E1:37_2_3_1_1"/><i><font color="Green" title="E63">A34</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">N2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> ex <font color="Olive" title="b2">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ex <font color="Olive" title="b3">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N2</font> &amp; <font color="Olive" title="b1">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Olive" title="b3">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">N2</font>,<font color="Olive" title="b2">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">Fn</font> &amp; <font color="Olive" title="b3">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><div class="add"><b>set </b><font color="Maroon" title="c19">S</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/><a NAME="E2:37_2_3_1_1"/><i><font color="Green" title="E64">A35</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:37_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="#E1:37_2_3_1_1"><i><font color="Green" title="E63">A34</font></i></a>, <a class="ref" href="xxreal_1.html#T17" title="XXREAL_1:th.17">XXREAL_1:17</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">S</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E4:37_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="#E1:37_2_3_1_1"><i><font color="Green" title="E63">A34</font></i></a>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="xxreal_1.html#T17" title="XXREAL_1:th.17">XXREAL_1:17</a></i>;<br/><b>take </b><font color="Maroon" title="c21">N2</font> = <font color="Maroon" title="c17">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ex <font color="Olive" title="b2">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c21">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N2</font> &amp; <font color="Maroon" title="c21">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Olive" title="b2">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Olive" title="b1">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">Fn</font> &amp; <font color="Olive" title="b2">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><a NAME="E4:37_2_3_1_1"/><i><font color="Green" title="E65">A36</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c21">N2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">N2</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E5:37_2_3_1_1"/><i><font color="Green" title="E66">A37</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">S</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E6:37_2_3_1_1"/><i><font color="Green" title="E67">A38</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c21">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c21">N2</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c22">Fn</font> = <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><a NAME="E7:37_2_3_1_1"/><i><font color="Green" title="E68">A39</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">Ft</font>
 <b>by </b><i><a class="txt" href="#E6:37"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="#E2:37_2_3_1_1"><i><font color="Green" title="E64">A35</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E8:37_2_3_1_1"/><i><font color="Green" title="E69">A40</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E6:37"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="#E2:37_2_3_1_1"><i><font color="Green" title="E64">A35</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E9:37_2_3_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c23">Fn</font> = <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c21">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="#E4:37_2_3_1_1"><i><font color="Green" title="E65">A36</font></i></a>, <a class="txt" href="#E5:37_2_3_1_1"><i><font color="Green" title="E66">A37</font></i></a>, <a class="txt" href="#E6:37_2_3_1_1"><i><font color="Green" title="E67">A38</font></i></a>, <a class="txt" href="#E8:37_2_3_1_1"><i><font color="Green" title="E69">A40</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><b>take </b>
<font color="Maroon" title="c20">S</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c21">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Maroon" title="c20">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N2</font> &amp; <font color="Maroon" title="c21">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Olive" title="b1">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">Fn</font> &amp; <font color="Olive" title="b1">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><b>take </b>
<font color="Maroon" title="c23">Fn</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N2</font> &amp; <font color="Maroon" title="c21">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Maroon" title="c23">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">Fn</font> &amp; <font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><b>thus </b><a NAME="E11:37_2_3_1_1"/>
<font color="Maroon" title="c20">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N2</font> &amp; <font color="Maroon" title="c21">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Maroon" title="c23">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">Fn</font> &amp; <font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><b>thus </b><a NAME="E12:37_2_3_1_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N2</font>
 <b>by </b><i><a class="txt" href="#E9:37_2"><i><font color="Green" title="E53">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Maroon" title="c23">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">Fn</font> &amp; <font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><b>thus </b><a NAME="E13:37_2_3_1_1"/>
<font color="Maroon" title="c21">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">Fn</font> &amp; <font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><b>reconsider </b><font color="Maroon" title="c24">K0</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="txt" href="#E5:37"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="#E2:37_2_3_1_1"><i><font color="Green" title="E64">A35</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">S1</font> = <font color="Maroon" title="c20">S</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span>)</span> <b>by </b><i><a class="txt" href="#E4:37_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="#E1:37_2_3_1_1"><i><font color="Green" title="E63">A34</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="ref" href="xxreal_1.html#T17" title="XXREAL_1:th.17">XXREAL_1:17</a></i>;<br/><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font> = 
 <a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> 
<b>by </b><i><a class="txt" href="#E2:37_2_3_1_1"><i><font color="Green" title="E64">A35</font></i></a>, <a class="ref" href="topalg_3.html#T6" title="TOPALG_3:th.6">TOPALG_3:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">S1</font>
<b>by </b><i><a class="txt" href="#E2:37_2_3_1_1"><i><font color="Green" title="E64">A35</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="ref" href="tsep_1.html#T3" title="TSEP_1:th.3">TSEP_1:3</a></i>
;<br/><a NAME="E17:37_2_3_1_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c21">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">K0</font>
 <b>by </b><i><a class="ref" href="borsuk_3.html#T26" title="BORSUK_3:th.26">BORSUK_3:26</a></i>;<br/><b>hence </b><a NAME="E18:37_2_3_1_1"/>
<font color="Maroon" title="c23">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E2:37"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="#E2:37_2_3_1_1"><i><font color="Green" title="E64">A35</font></i></a>, <a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">Fn</font> &amp; <font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><a NAME="E19:37_2_3_1_1"/><i><font color="Green" title="E70">A41</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E20:37_2_3_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c23">Fn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green" title="E29">Lm12</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><a NAME="E21:37_2_3_1_1"/><b>then </b><i><font color="Green" title="E71">A42</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">Fn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">Fn</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><a NAME="E22:37_2_3_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">Fn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">Fn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_3_1_1_2"/><i><font color="Green" title="E72">A43</font></i>: 
<font color="Maroon" title="c26">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">Fn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font></span><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font> = 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="#E19:37_2_3_1_1"><i><font color="Green" title="E70">A41</font></i></a>, <a class="txt" href="#E1:37_2_3_1_1_2"><i><font color="Green" title="E72">A43</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">Ft</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="#E3:37"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="#E5:37"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="#E8:37_2_3_1_1"><i><font color="Green" title="E69">A40</font></i></a>, <a class="txt" href="#E19:37_2_3_1_1"><i><font color="Green" title="E70">A41</font></i></a>, <a class="txt" href="#E1:37_2_3_1_1_2"><i><font color="Green" title="E72">A43</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:37_2_3_1_1"><i><font color="Green" title="E68">A39</font></i></a>, <a class="txt" href="#E19:37_2_3_1_1"><i><font color="Green" title="E70">A41</font></i></a>, <a class="txt" href="#E1:37_2_3_1_1_2"><i><font color="Green" title="E72">A43</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:37_2_3_1_1"><i><font color="Green" title="E64">A35</font></i></a>, <a class="txt" href="#E19:37_2_3_1_1"><i><font color="Green" title="E70">A41</font></i></a>, <a class="txt" href="#E1:37_2_3_1_1_2"><i><font color="Green" title="E72">A43</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">Fn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="#E8:37_2_3_1_1"><i><font color="Green" title="E69">A40</font></i></a>, <a class="txt" href="#E19:37_2_3_1_1"><i><font color="Green" title="E70">A41</font></i></a>, <a class="txt" href="#E1:37_2_3_1_1_2"><i><font color="Green" title="E72">A43</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E23:37_2_3_1_1"/>
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">Fn</font>
 <b>by </b><i><a class="txt" href="#E8:37_2_3_1_1"><i><font color="Green" title="E69">A40</font></i></a>, <a class="txt" href="#E19:37_2_3_1_1"><i><font color="Green" title="E70">A41</font></i></a>, <a class="txt" href="#E21:37_2_3_1_1"><i><font color="Green" title="E71">A42</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><a NAME="E24:37_2_3_1_1"/><i><font color="Green" title="E72">A44</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E8:37_2_3_1_1"><i><font color="Green" title="E69">A40</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E25:37_2_3_1_1"/><i><font color="Green" title="E73">A45</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E2:37_2_3_1_1"><i><font color="Green" title="E64">A35</font></i></a>, <a class="ref" href="zfmisc_1.html#T124" title="ZFMISC_1:th.124">ZFMISC_1:124</a></i>;<br/><i><font color="Green" title="E74">A46</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E6:37"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
<b>by </b><i><a class="txt" href="#E2:37_2_3_1_1"><i><font color="Green" title="E64">A35</font></i></a>, <a class="ref" href="zfmisc_1.html#T124" title="ZFMISC_1:th.124">ZFMISC_1:124</a></i>
;<br/><a NAME="E27:37_2_3_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_3_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_3_1_1_4"/><i><font color="Green" title="E75">A47</font></i>: 
<font color="Maroon" title="c26">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font></span><br/>

<a NAME="E2:37_2_3_1_1_4"/><i><font color="Green" title="E76">A48</font></i>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E2:37_2_3_1_1"><i><font color="Green" title="E64">A35</font></i></a>, <a class="txt" href="#E24:37_2_3_1_1"><i><font color="Green" title="E72">A44</font></i></a>, <a class="txt" href="#E1:37_2_3_1_1_4"><i><font color="Green" title="E75">A47</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:37_2_3_1_1_4"/><i><font color="Green" title="E77">A49</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E4:37_2_3_1_1_4"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E24:37_2_3_1_1"><i><font color="Green" title="E72">A44</font></i></a>, <a class="txt" href="#E1:37_2_3_1_1_4"><i><font color="Green" title="E75">A47</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font> = 
<font color="Maroon" title="c23">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c10">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="#E2:37_2_3_1_1_4"><i><font color="Green" title="E76">A48</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="#E2:37_2_3_1_1_4"><i><font color="Green" title="E76">A48</font></i></a>, <a class="txt" href="#E3:37_2_3_1_1_4"><i><font color="Green" title="E77">A49</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E28:37_2_3_1_1"/>
<font color="Maroon" title="c23">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E24:37_2_3_1_1"><i><font color="Green" title="E72">A44</font></i></a>, <a class="txt" href="#E25:37_2_3_1_1"><i><font color="Green" title="E73">A45</font></i></a>, <a class="txt" href="#E26:37_2_3_1_1"><i><font color="Green" title="E74">A46</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_3_1_2"><b>suppose </b></a><a NAME="E1:37_2_3_1_2"/><i><font color="Green" title="E63">A50</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">N2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> ex <font color="Olive" title="b2">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ex <font color="Olive" title="b3">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N2</font> &amp; <font color="Olive" title="b1">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Olive" title="b3">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">N2</font>,<font color="Olive" title="b2">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">Fn</font> &amp; <font color="Olive" title="b3">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c19">N2</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c20">S</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> , <font color="Maroon" title="c21">Fn</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <b> such that </b><br/><a NAME="E3:37_2_3_1_2"/><i><font color="Green" title="E64">A51</font></i>: 
<font color="Maroon" title="c20">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>and </b><br/><a NAME="E4:37_2_3_1_2"/><i><font color="Green" title="E65">A52</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">N2</font>
 <b>and </b><br/><a NAME="E5:37_2_3_1_2"/><i><font color="Green" title="E66">A53</font></i>: 
<font color="Maroon" title="c19">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font>
 <b>and </b><br/><a NAME="E6:37_2_3_1_2"/><i><font color="Green" title="E67">A54</font></i>: 
<font color="Maroon" title="c21">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E7:37_2_3_1_2"/><i><font color="Green" title="E68">A55</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">Fn</font>
 <b>and </b><br/><a NAME="E8:37_2_3_1_2"/><i><font color="Green" title="E69">A56</font></i>: 
<font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c19">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E1:37_2_3"><i><font color="Green" title="E61">A32</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">N2</font> = <font color="Maroon" title="c19">N2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> <b>by </b><i><a class="txt" href="#E4:37_2_3_1_2"><i><font color="Green" title="E65">A52</font></i></a></i>;<br/><a NAME="E10:37_2_3_1_2"/><i><font color="Green" title="E70">A57</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">S</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E11:37_2_3_1_2"/><i><font color="Green" title="E71">A58</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c23">SS</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/><b>consider </b><font color="Maroon" title="c24">Ui</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E13:37_2_3_1_2"/><i><font color="Green" title="E72">A59</font></i>: 
<font color="Maroon" title="c24">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">UL</font>
 <b>and </b><br/><a NAME="E14:37_2_3_1_2"/><i><font color="Green" title="E73">A60</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">Ui</font>
 <b>by </b><i><a class="txt" href="#E10:37_2"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E2:37_2_3"><i><font color="Green" title="E62">A33</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2"><i><font color="Green" title="E63">A50</font></i></a></i>;<br/><a NAME="E15:37_2_3_1_2"/><i><font color="Green" title="E74">A61</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">Ui</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">Ui</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><b>consider </b><font color="Maroon" title="c25">D</font> being  <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <b> such that </b><br/><a NAME="E17:37_2_3_1_2"/><i><font color="Green" title="E75">A62</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c25">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">Ui</font>
 <b>and </b><br/><a NAME="E18:37_2_3_1_2"/><i><font color="Green" title="E76">A63</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">D</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">Ui</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Olive" title="b2">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 <b>by </b><i><a class="txt" href="#E10:37"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="#E13:37_2_3_1_2"><i><font color="Green" title="E72">A59</font></i></a></i>;<br/><a NAME="E19:37_2_3_1_2"/><i><font color="Green" title="E77">A64</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">N2</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E20:37_2_3_1_2"/><i><font color="Green" title="E78">A65</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="#E14:37_2"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2"><i><font color="Green" title="E63">A50</font></i></a></i>;<br/><a NAME="E21:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E79">A66</font></i>: 
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">S</font>
 <b>by </b><i><a class="txt" href="#E3:37_2_3_1_2"><i><font color="Green" title="E64">A51</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E22:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E80">A67</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E4:37_2_3_1_2"><i><font color="Green" title="E65">A52</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then </b><i><font color="Green" title="E81">A68</font></i>: <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">Fn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="#E7:37_2_3_1_2"><i><font color="Green" title="E68">A55</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E11:37_2_3_1_2"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="#E19:37_2_3_1_2"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="#E22:37_2_3_1_2"><i><font color="Green" title="E80">A67</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/><a NAME="E24:37_2_3_1_2"/><i><font color="Green" title="E82">A69</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <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="c9">F</font>
 <b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E21:37_2_3_1_2"><i><font color="Green" title="E79">A66</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E25:37_2_3_1_2"/><i><font color="Green" title="E83">A70</font></i>: 
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E14:37_2"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="#E2:37_2_3"><i><font color="Green" title="E62">A33</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2"><i><font color="Green" title="E63">A50</font></i></a></i>;<br/><a NAME="E26:37_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E27:37_2_3_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E9:37_2"><i><font color="Green" title="E53">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E28:37_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E24:37_2_3_1_2"><i><font color="Green" title="E82">A69</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/><a NAME="E29:37_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">Ui</font>
 <b>by </b><i><a class="txt" href="#E14:37_2_3_1_2"><i><font color="Green" title="E73">A60</font></i></a>, <a class="txt" href="#E23:37_2_3_1_2"><i><font color="Green" title="E81">A68</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">Uit</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E31:37_2_3_1_2"/><i><font color="Green" title="E84">A71</font></i>: 
<font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">Uit</font>
 <b>and </b><br/><a NAME="E32:37_2_3_1_2"/><i><font color="Green" title="E85">A72</font></i>: 
<font color="Maroon" title="c26">Uit</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">D</font>
 <b>by </b><i><a class="txt" href="#E17:37_2_3_1_2"><i><font color="Green" title="E75">A62</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">Uit</font> = <font color="Maroon" title="c26">Uit</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="#E31:37_2_3_1_2"><i><font color="Green" title="E84">A71</font></i></a>, <a class="txt" href="#E32:37_2_3_1_2"><i><font color="Green" title="E85">A72</font></i></a></i>;<br/><a NAME="E34:37_2_3_1_2"/><i><font color="Green" title="E86">A73</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">Fn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E35:37_2_3_1_2"/><i><font color="Green" title="E87">A74</font></i>: 
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E36:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E88">A75</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E4:37_2_3_1_2"><i><font color="Green" title="E65">A52</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E37:37_2_3_1_2"/><i><font color="Green" title="E89">A76</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">S</font>
 <b>by </b><i><a class="txt" href="#E21:37_2_3_1_2"><i><font color="Green" title="E79">A66</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">Ti</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E21:37_2_3_1_2"><i><font color="Green" title="E79">A66</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E39:37_2_3_1_2"/><i><font color="Green" title="E90">A77</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c28">Ti</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Ti</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E40:37_2_3_1_2"/><i><font color="Green" title="E91">A78</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c28">Ti</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c28">Ti</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><b>set </b><font color="Maroon" title="c29">FnT</font> = <font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<font color="Maroon" title="c28">Ti</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/><a NAME="E41:37_2_3_1_2"/><i><font color="Green" title="E92">A79</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<font color="Maroon" title="c28">Ti</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E11:37_2_3_1_2"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="#E19:37_2_3_1_2"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="#E34:37_2_3_1_2"><i><font color="Green" title="E86">A73</font></i></a>, <a class="txt" href="#E37:37_2_3_1_2"><i><font color="Green" title="E89">A76</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E42:37_2_3_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<font color="Maroon" title="c28">Ti</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c30">FnT</font> = <font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<font color="Maroon" title="c28">Ti</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c28">Ti</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="#E19:37_2_3_1_2"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="#E39:37_2_3_1_2"><i><font color="Green" title="E90">A77</font></i></a>, <a class="txt" href="#E40:37_2_3_1_2"><i><font color="Green" title="E91">A78</font></i></a>, <a class="txt" href="#E41:37_2_3_1_2"><i><font color="Green" title="E92">A79</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E44:37_2_3_1_2"/>
<font color="Maroon" title="c22">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">N2</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c31">N7</font> = <font color="Maroon" title="c22">N2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c32">Ti2</font> = <font color="Maroon" title="c28">Ti</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span> <b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E21:37_2_3_1_2"><i><font color="Green" title="E79">A66</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E47:37_2_3_1_2"/><i><font color="Green" title="E93">A80</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">N7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font> &amp; <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c32">Ti2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c28">Ti</font> )
 <b>by </b><i><a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a></i>;<br/><a NAME="E48:37_2_3_1_2"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c31">N7</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c32">Ti2</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">N7</font>,<font color="Maroon" title="c32">Ti2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_3.html#T26" title="BORSUK_3:th.26">BORSUK_3:26</a></i>;<br/><a NAME="E49:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E94">A81</font></i>: 
<font color="Maroon" title="c30">FnT</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E6:37_2_3_1_2"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="#E47:37_2_3_1_2"><i><font color="Green" title="E93">A80</font></i></a>, <a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/><a NAME="E50:37_2_3_1_2"/><i><font color="Green" title="E95">A82</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E51:37_2_3_1_2"/>
<font color="Maroon" title="c27">Uit</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E32:37_2_3_1_2"><i><font color="Green" title="E85">A72</font></i></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/><a NAME="E52:37_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c30">FnT</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c27">Uit</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E49:37_2_3_1_2"><i><font color="Green" title="E94">A81</font></i></a>, <a class="txt" href="#E50:37_2_3_1_2"><i><font color="Green" title="E95">A82</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/><b>then consider </b><font color="Maroon" title="c33">SF</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c28">Ti</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E54:37_2_3_1_2"/><i><font color="Green" title="E96">A83</font></i>: 
<font color="Maroon" title="c30">FnT</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c27">Uit</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c33">SF</font>
 <b>and </b><br/><a NAME="E55:37_2_3_1_2"/><i><font color="Green" title="E97">A84</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">SF</font> holds <br/> ex <font color="Olive" title="b2">X1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span> ex <font color="Olive" title="b3">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c28">Ti</font></span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <font color="Olive" title="b2">X1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">Y1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T45" title="BORSUK_1:th.45">BORSUK_1:45</a></i>;<br/><a NAME="E56:37_2_3_1_2"/>
<font color="Maroon" title="c30">FnT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">Uit</font>
 <b>by </b><i><a class="txt" href="#E31:37_2_3_1_2"><i><font color="Green" title="E84">A71</font></i></a>, <a class="txt" href="#E36:37_2_3_1_2"><i><font color="Green" title="E88">A75</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E57:37_2_3_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">FnT</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c27">Uit</font>
 <b>by </b><i><a class="txt" href="#E36:37_2_3_1_2"><i><font color="Green" title="E88">A75</font></i></a>, <a class="txt" href="#E41:37_2_3_1_2"><i><font color="Green" title="E92">A79</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c34">N5</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E59:37_2_3_1_2"/><i><font color="Green" title="E98">A85</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">N5</font>
 <b>and </b><br/><a NAME="E60:37_2_3_1_2"/><i><font color="Green" title="E99">A86</font></i>: 
<font color="Maroon" title="c34">N5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">SF</font>
 <b>by </b><i><a class="txt" href="#E54:37_2_3_1_2"><i><font color="Green" title="E96">A83</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c35">X1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>, <font color="Maroon" title="c36">Y1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c28">Ti</font></span>)</span><b> such that </b><br/><a NAME="E62:37_2_3_1_2"/><i><font color="Green" title="E100">A87</font></i>: 
<font color="Maroon" title="c34">N5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c35">X1</font>,<font color="Maroon" title="c36">Y1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E63:37_2_3_1_2"/><i><font color="Green" title="E101">A88</font></i>: 
<font color="Maroon" title="c35">X1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E64:37_2_3_1_2"/>
<font color="Maroon" title="c36">Y1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E55:37_2_3_1_2"><i><font color="Green" title="E97">A84</font></i></a>, <a class="txt" href="#E60:37_2_3_1_2"><i><font color="Green" title="E99">A86</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c37">NY</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font><b> such that </b><br/><a NAME="E66:37_2_3_1_2"/><i><font color="Green" title="E102">A89</font></i>: 
<font color="Maroon" title="c37">NY</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E67:37_2_3_1_2"/><i><font color="Green" title="E103">A90</font></i>: 
<font color="Maroon" title="c37">NY</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">X1</font>
 <b>by </b><i><a class="txt" href="#E63:37_2_3_1_2"><i><font color="Green" title="E101">A88</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/><b>consider </b><font color="Maroon" title="c38">y1</font>, <font color="Maroon" title="c39">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E69:37_2_3_1_2"/><i><font color="Green" title="E104">A91</font></i>: 
<font color="Maroon" title="c38">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">X1</font>
 <b>and </b><br/><a NAME="E70:37_2_3_1_2"/><i><font color="Green" title="E105">A92</font></i>: 
<font color="Maroon" title="c39">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Y1</font>
 <b>and </b><br/><a NAME="E71:37_2_3_1_2"/><i><font color="Green" title="E106">A93</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c38">y1</font>,<font color="Maroon" title="c39">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E59:37_2_3_1_2"><i><font color="Green" title="E98">A85</font></i></a>, <a class="txt" href="#E62:37_2_3_1_2"><i><font color="Green" title="E100">A87</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>set </b><font color="Maroon" title="c40">N1</font> = <font color="Maroon" title="c37">NY</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c22">N2</font>;<br/><a NAME="E72:37_2_3_1_2"/><i><font color="Green" title="E107">A94</font></i>: 
<font color="Maroon" title="c36">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">Ti</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_3_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_2_3_1_2_2"/>
<font color="Maroon" title="c36">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c28">Ti</font>
 <b>by </b><i><a class="txt" href="#E39:37_2_3_1_2"><i><font color="Green" title="E90">A77</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c28">Ti</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c36">Y1</font></span><br/>

<b>let </b><font color="Maroon" title="c41">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">Ti</font> or <font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Y1</font> )</span><br/>

<b>assume </b><a NAME="E2:37_2_3_1_2_2"/>
<font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">Ti</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Y1</font></span><br/>

<a NAME="E3:37_2_3_1_2_2"/><b>then </b>
<font color="Maroon" title="c41">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:37_2_3_1_2_2"/>
<font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Y1</font>
 <b>by </b><i><a class="txt" href="#E70:37_2_3_1_2"><i><font color="Green" title="E105">A92</font></i></a>, <a class="txt" href="#E71:37_2_3_1_2"><i><font color="Green" title="E106">A93</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E73:37_2_3_1_2"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">y1</font>
 <b>by </b><i><a class="txt" href="#E71:37_2_3_1_2"><i><font color="Green" title="E106">A93</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E74:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E108">A95</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">NY</font>
 <b>by </b><i><a class="txt" href="#E67:37_2_3_1_2"><i><font color="Green" title="E103">A90</font></i></a>, <a class="txt" href="#E69:37_2_3_1_2"><i><font color="Green" title="E104">A91</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c41">N1</font> = <font color="Maroon" title="c37">NY</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c22">N2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> <b>by </b><i><a class="txt" href="#E4:37_2_3_1_2"><i><font color="Green" title="E65">A52</font></i></a>, <a class="txt" href="#E66:37_2_3_1_2"><i><font color="Green" title="E102">A89</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E76:37_2_3_1_2"/><i><font color="Green" title="E109">A96</font></i>: 
<font color="Maroon" title="c41">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">N2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E77:37_2_3_1_2"/><i><font color="Green" title="E110">A97</font></i>: 
<font color="Maroon" title="c21">Fn</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">Uit</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_3_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c42">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">Fn</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">Uit</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_3_1_2_3"/>
<font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">Fn</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">Uit</font></span><br/>

<b>then consider </b><font color="Maroon" title="c43">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E3:37_2_3_1_2_3"/><i><font color="Green" title="E111">A98</font></i>: 
<font color="Maroon" title="c43">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E4:37_2_3_1_2_3"/><i><font color="Green" title="E112">A99</font></i>: 
<font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c43">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:37_2_3_1_2_3"/>
<font color="Maroon" title="c43">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">N5</font>
 
<b>by </b><i><a class="txt" href="#E62:37_2_3_1_2"><i><font color="Green" title="E100">A87</font></i></a>, <a class="txt" href="#E67:37_2_3_1_2"><i><font color="Green" title="E103">A90</font></i></a>, <a class="txt" href="#E72:37_2_3_1_2"><i><font color="Green" title="E107">A94</font></i></a>, <a class="txt" href="#E3:37_2_3_1_2_3"><i><font color="Green" title="E111">A98</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E6:37_2_3_1_2_3"/><b>then </b><i><font color="Green" title="E113">A100</font></i>: 
<font color="Maroon" title="c43">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c33">SF</font>
 
<b>by </b><i><a class="txt" href="#E60:37_2_3_1_2"><i><font color="Green" title="E99">A86</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:37_2_3_1_2_3"/><b>then </b>
<font color="Maroon" title="c43">a</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="c30">FnT</font>
 
<b>by </b><i><a class="txt" href="#E54:37_2_3_1_2"><i><font color="Green" title="E96">A83</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E8:37_2_3_1_2_3"/><b>then </b>
<font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c43">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">FnT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c43">a</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E9:37_2_3_1_2_3"/>
<font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">Uit</font>
 <b>by </b><i><a class="txt" href="#E54:37_2_3_1_2"><i><font color="Green" title="E96">A83</font></i></a>, <a class="txt" href="#E4:37_2_3_1_2_3"><i><font color="Green" title="E112">A99</font></i></a>, <a class="txt" href="#E6:37_2_3_1_2_3"><i><font color="Green" title="E113">A100</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E78:37_2_3_1_2"/><i><font color="Green" title="E111">A101</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">N1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E79:37_2_3_1_2"/><i><font color="Green" title="E112">A102</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E5:37_2_3_1_2"><i><font color="Green" title="E66">A53</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E80:37_2_3_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E76:37_2_3_1_2"><i><font color="Green" title="E109">A96</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E81:37_2_3_1_2"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E79:37_2_3_1_2"><i><font color="Green" title="E112">A102</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E82:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E113">A103</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/><b>set </b><font color="Maroon" title="c42">f</font> = <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">Uit</font>;<br/><a NAME="E83:37_2_3_1_2"/><i><font color="Green" title="E114">A104</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c27">Uit</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">Uit</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E84:37_2_3_1_2"/><i><font color="Green" title="E115">A105</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">Uit</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">Uit</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green" title="E29">Lm12</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><a NAME="E85:37_2_3_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">Uit</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">Ui</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_3_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c43">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c43">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">Uit</font></span>)</span> or <font color="Maroon" title="c43">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Ui</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_3_1_2_4"/>
<font color="Maroon" title="c43">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">Uit</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c43">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Ui</font></span><br/>

<b>then consider </b><font color="Maroon" title="c44">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_2_3_1_2_4"/><i><font color="Green" title="E116">A106</font></i>: 
<font color="Maroon" title="c44">a</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> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">Uit</font></span>)</span>
 <b>and </b><br/><a NAME="E4:37_2_3_1_2_4"/><i><font color="Green" title="E117">A107</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">Uit</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c43">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:37_2_3_1_2_4"/>
<font color="Maroon" title="c44">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c25">D</font>
 
<b>by </b><i><a class="txt" href="#E32:37_2_3_1_2"><i><font color="Green" title="E85">A72</font></i></a>, <a class="txt" href="#E84:37_2_3_1_2"><i><font color="Green" title="E115">A105</font></i></a>, <a class="txt" href="#E3:37_2_3_1_2_4"><i><font color="Green" title="E116">A106</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:37_2_3_1_2_4"/><b>then </b>
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Ui</font>
 
<b>by </b><i><a class="txt" href="#E17:37_2_3_1_2"><i><font color="Green" title="E75">A62</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E7:37_2_3_1_2_4"/>
<font color="Maroon" title="c43">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Ui</font>
 <b>by </b><i><a class="txt" href="#E84:37_2_3_1_2"><i><font color="Green" title="E115">A105</font></i></a>, <a class="txt" href="#E3:37_2_3_1_2_4"><i><font color="Green" title="E116">A106</font></i></a>, <a class="txt" href="#E4:37_2_3_1_2_4"><i><font color="Green" title="E117">A107</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c43">f</font> = <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">Uit</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c27">Uit</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">Ui</font></span>)</span> <b>by </b><i><a class="txt" href="#E15:37_2_3_1_2"><i><font color="Green" title="E74">A61</font></i></a>, <a class="txt" href="#E83:37_2_3_1_2"><i><font color="Green" title="E114">A104</font></i></a>, <a class="txt" href="#E84:37_2_3_1_2"><i><font color="Green" title="E115">A105</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E87:37_2_3_1_2"/><i><font color="Green" title="E116">A108</font></i>: 
<font color="Maroon" title="c43">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 <b>by </b><i><a class="txt" href="#E18:37_2_3_1_2"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="#E32:37_2_3_1_2"><i><font color="Green" title="E85">A72</font></i></a></i>;<br/><a NAME="E88:37_2_3_1_2"/>
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="#E14:37_2"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="#E2:37_2_3"><i><font color="Green" title="E62">A33</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2"><i><font color="Green" title="E63">A50</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c44">SS</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E15:37_2"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="#E20:37_2_3_1_2"><i><font color="Green" title="E78">A65</font></i></a>, <a class="txt" href="#E25:37_2_3_1_2"><i><font color="Green" title="E83">A70</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c41">N1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ex <font color="Olive" title="b2">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">N1</font> &amp; <font color="Maroon" title="c41">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Olive" title="b2">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Olive" title="b1">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">Fn</font> &amp; <font color="Olive" title="b2">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><b>take </b><font color="Maroon" title="c45">S1</font> = <font color="Maroon" title="c20">S</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c44">SS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c45">S1</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Maroon" title="c45">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">N1</font> &amp; <font color="Maroon" title="c41">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Olive" title="b1">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c45">S1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">Fn</font> &amp; <font color="Olive" title="b1">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><b>set </b><font color="Maroon" title="c46">Fni1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c47">ff</font> = <font color="Maroon" title="c43">f</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><a NAME="E91:37_2_3_1_2"/><i><font color="Green" title="E117">A109</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">F</font>
 <b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E92:37_2_3_1_2"/>
<font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 <b>by </b><i><a class="txt" href="#E18:37_2_3_1_2"><i><font color="Green" title="E76">A63</font></i></a>, <a class="txt" href="#E32:37_2_3_1_2"><i><font color="Green" title="E85">A72</font></i></a>, <a class="ref" href="tops_2.html#T70" title="TOPS_2:th.70">TOPS_2:70</a></i>;<br/><a NAME="E93:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E118">A110</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">Ui</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E94:37_2_3_1_2"/><i><font color="Green" title="E119">A111</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c43">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">Ui</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E87:37_2_3_1_2"><i><font color="Green" title="E116">A108</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E95:37_2_3_1_2"/><i><font color="Green" title="E120">A112</font></i>: 
<font color="Maroon" title="c43">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="#E87:37_2_3_1_2"><i><font color="Green" title="E116">A108</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E96:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E121">A113</font></i>: 
<font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="#E94:37_2_3_1_2"><i><font color="Green" title="E119">A111</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/><a NAME="E97:37_2_3_1_2"/><i><font color="Green" title="E122">A114</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c44">SS</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E98:37_2_3_1_2"/><i><font color="Green" title="E123">A115</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c45">S1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">S1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E99:37_2_3_1_2"/><i><font color="Green" title="E124">A116</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E100:37_2_3_1_2"/><i><font color="Green" title="E125">A117</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_3_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c48">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c48">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> or <font color="Maroon" title="c48">b</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> <span class="p1">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:37_2_3_1_2_5"/>
<font color="Maroon" title="c48">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c48">b</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> <span class="p1">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c49">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_2_3_1_2_5"/><i><font color="Green" title="E126">A118</font></i>: 
<font color="Maroon" title="c49">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:37_2_3_1_2_5"/><i><font color="Green" title="E127">A119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:37_2_3_1_2_5"/>
<font color="Maroon" title="c48">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">a</font>
 
<b>by </b><i><a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E3:37_2_3_1_2_5"><i><font color="Green" title="E126">A118</font></i></a>, <a class="txt" href="#E4:37_2_3_1_2_5"><i><font color="Green" title="E127">A119</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:37_2_3_1_2_5"/><b>then </b>
<font color="Maroon" title="c48">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E3:37_2_3_1_2_5"><i><font color="Green" title="E126">A118</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E7:37_2_3_1_2_5"/><b>then </b>
<font color="Maroon" title="c48">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E82:37_2_3_1_2"><i><font color="Green" title="E113">A103</font></i></a></i>;<br/>
<a NAME="E8:37_2_3_1_2_5"/><b>then </b>
<font color="Maroon" title="c48">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Ui</font>
 
<b>by </b><i><a class="txt" href="#E14:37_2_3_1_2"><i><font color="Green" title="E73">A60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:37_2_3_1_2_5"/>
<font color="Maroon" title="c48">b</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> <span class="p1">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E93:37_2_3_1_2"><i><font color="Green" title="E118">A110</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E101:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E126">A120</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><a NAME="E102:37_2_3_1_2"/><i><font color="Green" title="E127">A121</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="txt" href="#E78:37_2_3_1_2"><i><font color="Green" title="E111">A101</font></i></a>, <a class="txt" href="#E97:37_2_3_1_2"><i><font color="Green" title="E122">A114</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><a NAME="E103:37_2_3_1_2"/><i><font color="Green" title="E128">A122</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c45">S1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c45">S1</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="txt" href="#E78:37_2_3_1_2"><i><font color="Green" title="E111">A101</font></i></a>, <a class="txt" href="#E98:37_2_3_1_2"><i><font color="Green" title="E123">A115</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><a NAME="E104:37_2_3_1_2"/><i><font color="Green" title="E129">A123</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E78:37_2_3_1_2"><i><font color="Green" title="E111">A101</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/><a NAME="E105:37_2_3_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c27">Uit</font></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/><a NAME="E106:37_2_3_1_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c48">Fni1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E101:37_2_3_1_2"><i><font color="Green" title="E126">A120</font></i></a>, <a class="txt" href="#E102:37_2_3_1_2"><i><font color="Green" title="E127">A121</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><b>set </b><font color="Maroon" title="c49">Fn2</font> = <font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/><b>set </b><font color="Maroon" title="c50">Fn1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c48">Fni1</font>;<br/><a NAME="E108:37_2_3_1_2"/><i><font color="Green" title="E130">A124</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c45">S1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T120" title="ZFMISC_1:th.120">ZFMISC_1:120</a></i>;<br/><a NAME="E109:37_2_3_1_2"/><i><font color="Green" title="E131">A125</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E76:37_2_3_1_2"><i><font color="Green" title="E109">A96</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E110:37_2_3_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E11:37_2_3_1_2"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="#E19:37_2_3_1_2"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="#E34:37_2_3_1_2"><i><font color="Green" title="E86">A73</font></i></a>, <a class="txt" href="#E76:37_2_3_1_2"><i><font color="Green" title="E109">A96</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E111:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E132">A126</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c48">Fni1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E101:37_2_3_1_2"><i><font color="Green" title="E126">A120</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/><a NAME="E112:37_2_3_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c48">Fni1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c48">Fni1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c51">Fn1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c48">Fni1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c45">S1</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="#E103:37_2_3_1_2"><i><font color="Green" title="E128">A122</font></i></a>, <a class="txt" href="#E108:37_2_3_1_2"><i><font color="Green" title="E130">A124</font></i></a>, <a class="txt" href="#E111:37_2_3_1_2"><i><font color="Green" title="E132">A126</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c51">Fn1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c45">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">N1</font> &amp; <font color="Maroon" title="c41">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Maroon" title="c51">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c45">S1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font> &amp; <font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><b>thus </b><a NAME="E114:37_2_3_1_2"/><i><font color="Green" title="E133">A127</font></i>: 
<font color="Maroon" title="c45">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="#E3:37_2_3_1_2"><i><font color="Green" title="E64">A51</font></i></a>, <a class="txt" href="#E20:37_2_3_1_2"><i><font color="Green" title="E78">A65</font></i></a>, <a class="txt" href="#E25:37_2_3_1_2"><i><font color="Green" title="E83">A70</font></i></a>, <a class="ref" href="xxreal_1.html#T165" title="XXREAL_1:th.165">XXREAL_1:165</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">N1</font> &amp; <font color="Maroon" title="c41">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Maroon" title="c51">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c45">S1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font> &amp; <font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><b>thus </b><a NAME="E115:37_2_3_1_2"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">N1</font>
 <b>by </b><i><a class="txt" href="#E4:37_2_3_1_2"><i><font color="Green" title="E65">A52</font></i></a>, <a class="txt" href="#E74:37_2_3_1_2"><i><font color="Green" title="E108">A95</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c41">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font> &amp; <font color="Maroon" title="c51">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c45">S1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font> &amp; <font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><b>thus </b><a NAME="E116:37_2_3_1_2"/>
<font color="Maroon" title="c41">N1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font>
 <b>by </b><i><a class="txt" href="#E5:37_2_3_1_2"><i><font color="Green" title="E66">A53</font></i></a>, <a class="txt" href="#E76:37_2_3_1_2"><i><font color="Green" title="E109">A96</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c51">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c45">S1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font> &amp; <font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><a NAME="E117:37_2_3_1_2"/><i><font color="Green" title="E134">A128</font></i>: 
<font color="Maroon" title="c20">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c45">S1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E118:37_2_3_1_2"/><i><font color="Green" title="E135">A129</font></i>: 
<font color="Maroon" title="c44">SS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c45">S1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E119:37_2_3_1_2"/><i><font color="Green" title="E136">A130</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E11:37_2_3_1_2"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="#E19:37_2_3_1_2"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="#E34:37_2_3_1_2"><i><font color="Green" title="E86">A73</font></i></a>, <a class="txt" href="#E76:37_2_3_1_2"><i><font color="Green" title="E109">A96</font></i></a>, <a class="txt" href="#E104:37_2_3_1_2"><i><font color="Green" title="E129">A123</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E120:37_2_3_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c52">Fn2</font> = <font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="#E119:37_2_3_1_2"><i><font color="Green" title="E136">A130</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E122:37_2_3_1_2"/><i><font color="Green" title="E137">A131</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c45">S1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c45">S1</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c53">F1</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span>, <font color="Maroon" title="c54">F2</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c45">S1</font></span>)</span> <b>by </b><i><a class="txt" href="#E98:37_2_3_1_2"><i><font color="Green" title="E123">A115</font></i></a>, <a class="txt" href="#E117:37_2_3_1_2"><i><font color="Green" title="E134">A128</font></i></a>, <a class="txt" href="#E118:37_2_3_1_2"><i><font color="Green" title="E135">A129</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E124:37_2_3_1_2"/><i><font color="Green" title="E138">A132</font></i>: 
<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c45">S1</font>
 <b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E98:37_2_3_1_2"><i><font color="Green" title="E123">A115</font></i></a>, <a class="txt" href="#E117:37_2_3_1_2"><i><font color="Green" title="E134">A128</font></i></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/><a NAME="E125:37_2_3_1_2"/><i><font color="Green" title="E139">A133</font></i>: 
<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c45">S1</font>
 <b>by </b><i><a class="txt" href="#E97:37_2_3_1_2"><i><font color="Green" title="E122">A114</font></i></a>, <a class="txt" href="#E98:37_2_3_1_2"><i><font color="Green" title="E123">A115</font></i></a>, <a class="txt" href="#E118:37_2_3_1_2"><i><font color="Green" title="E135">A129</font></i></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c55">hS</font> = <font color="Maroon" title="c53">F1</font>, <font color="Maroon" title="c56">hSS</font> = <font color="Maroon" title="c54">F2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E127:37_2_3_1_2"/>
<font color="Maroon" title="c55">hS</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="#E3:37_2_3_1_2"><i><font color="Green" title="E64">A51</font></i></a>, <a class="ref" href="borsuk_4.html#T48" title="BORSUK_4:th.48">BORSUK_4:48</a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E128:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E140">A134</font></i>: 
<font color="Maroon" title="c53">F1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tsep_1.html#T8" title="TSEP_1:th.8">TSEP_1:8</a></i>;<br/><a NAME="E129:37_2_3_1_2"/>
<font color="Maroon" title="c56">hSS</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="borsuk_4.html#T48" title="BORSUK_4:th.48">BORSUK_4:48</a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E130:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E141">A135</font></i>: 
<font color="Maroon" title="c54">F2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tsep_1.html#T8" title="TSEP_1:th.8">TSEP_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c57">K0</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E11:37_2_3_1_2"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="#E109:37_2_3_1_2"><i><font color="Green" title="E131">A125</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c58">aN1</font> = <font color="Maroon" title="c41">N1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span> <b>by </b><i><a class="txt" href="#E76:37_2_3_1_2"><i><font color="Green" title="E109">A96</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E133:37_2_3_1_2"/>
<font color="Maroon" title="c20">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c59">aS</font> = <font color="Maroon" title="c20">S</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c57">K0</font> = 
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aN1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c59">aS</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_3.html#T26" title="BORSUK_3:th.26">BORSUK_3:26</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c59">aS</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
<b>by </b><i><a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
<b>by </b><i><a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a></i>
;<br/><a NAME="E136:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E142">A136</font></i>: 
<font color="Maroon" title="c52">Fn2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E6:37_2_3_1_2"><i><font color="Green" title="E67">A54</font></i></a>, <a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c60">fF</font> = <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c24">Ui</font></span>)</span> <b>by </b><i><a class="txt" href="#E93:37_2_3_1_2"><i><font color="Green" title="E118">A110</font></i></a>, <a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E100:37_2_3_1_2"><i><font color="Green" title="E125">A117</font></i></a>, <a class="txt" href="#E102:37_2_3_1_2"><i><font color="Green" title="E127">A121</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c61">gF</font> = <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E100:37_2_3_1_2"><i><font color="Green" title="E125">A117</font></i></a>, <a class="txt" href="#E102:37_2_3_1_2"><i><font color="Green" title="E127">A121</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E139:37_2_3_1_2"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_3.html#T26" title="BORSUK_3:th.26">BORSUK_3:26</a></i>;<br/><a NAME="E140:37_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c61">gF</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/><a NAME="E141:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E143">A137</font></i>: 
<font color="Maroon" title="c60">fF</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T9" title="TOPMETR:th.9">TOPMETR:9</a></i>;<br/><a NAME="E142:37_2_3_1_2"/>
<font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E87:37_2_3_1_2"><i><font color="Green" title="E116">A108</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E143:37_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c60">fF</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E141:37_2_3_1_2"><i><font color="Green" title="E143">A137</font></i></a></i>;<br/><a NAME="E144:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E144">A138</font></i>: 
<font color="Maroon" title="c48">Fni1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#T56" title="PRE_TOPC:th.56">PRE_TOPC:56</a></i>;<br/><a NAME="E145:37_2_3_1_2"/>
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span>)</span> holds <br/><font color="Maroon" title="c52">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">Fni1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_3_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c62">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c62">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span>)</span> implies <font color="Maroon" title="c52">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">Fni1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_3_1_2_7"/><i><font color="Green" title="E145">A139</font></i>: 
<font color="Maroon" title="c62">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c52">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">Fni1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font></span><br/>

<a NAME="E2:37_2_3_1_2_7"/><i><font color="Green" title="E146">A140</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c44">SS</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T122" title="ZFMISC_1:th.122">ZFMISC_1:122</a></i>;<br/>
<a NAME="E3:37_2_3_1_2_7"/><i><font color="Green" title="E147">A141</font></i>: 
<font color="Maroon" title="c20">S</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c44">SS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="#E3:37_2_3_1_2"><i><font color="Green" title="E64">A51</font></i></a>, <a class="txt" href="#E20:37_2_3_1_2"><i><font color="Green" title="E78">A65</font></i></a>, <a class="txt" href="#E25:37_2_3_1_2"><i><font color="Green" title="E83">A70</font></i></a>, <a class="ref" href="xxreal_1.html#T418" title="XXREAL_1:th.418">XXREAL_1:418</a></i>;<br/>
<a NAME="E4:37_2_3_1_2_7"/><i><font color="Green" title="E148">A142</font></i>: 
<font color="Maroon" title="c62">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c44">SS</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:37_2_3_1_2_7"><i><font color="Green" title="E145">A139</font></i></a></i>;<br/>
<a NAME="E5:37_2_3_1_2_7"/><i><font color="Green" title="E149">A143</font></i>: 
<font color="Maroon" title="c62">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:37_2_3_1_2"><i><font color="Green" title="E64">A51</font></i></a>, <a class="txt" href="#E20:37_2_3_1_2"><i><font color="Green" title="E78">A65</font></i></a>, <a class="txt" href="#E25:37_2_3_1_2"><i><font color="Green" title="E83">A70</font></i></a>, <a class="txt" href="#E102:37_2_3_1_2"><i><font color="Green" title="E127">A121</font></i></a>, <a class="txt" href="#E104:37_2_3_1_2"><i><font color="Green" title="E129">A123</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_7"><i><font color="Green" title="E145">A139</font></i></a>, <a class="txt" href="#E2:37_2_3_1_2_7"><i><font color="Green" title="E146">A140</font></i></a>, <a class="ref" href="xxreal_1.html#T418" title="XXREAL_1:th.418">XXREAL_1:418</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c63">p1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c41">N1</font>, <font color="Maroon" title="c64">p2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span><b> such that </b><br/><a NAME="E7:37_2_3_1_2_7"/><i><font color="Green" title="E150">A144</font></i>: 
<font color="Maroon" title="c62">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c63">p1</font>,<font color="Maroon" title="c64">p2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E8:37_2_3_1_2_7"/><i><font color="Green" title="E151">A145</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">N2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">N2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E9:37_2_3_1_2_7"/><i><font color="Green" title="E152">A146</font></i>: 
<font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font>
 
<b>by </b><i><a class="txt" href="#E104:37_2_3_1_2"><i><font color="Green" title="E129">A123</font></i></a>, <a class="txt" href="#E4:37_2_3_1_2_7"><i><font color="Green" title="E148">A142</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E10:37_2_3_1_2_7"/><i><font color="Green" title="E153">A147</font></i>: 
<font color="Maroon" title="c64">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">S</font>
 
<b>by </b><i><a class="txt" href="#E3:37_2_3_1_2_7"><i><font color="Green" title="E147">A141</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:37_2_3_1_2_7"/>
<font color="Maroon" title="c63">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">N1</font>
 
;<br/>
<a NAME="E12:37_2_3_1_2_7"/><b>then </b><i><font color="Green" title="E154">A148</font></i>: 
<font color="Maroon" title="c62">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E76:37_2_3_1_2"><i><font color="Green" title="E109">A96</font></i></a>, <a class="txt" href="#E7:37_2_3_1_2_7"><i><font color="Green" title="E150">A144</font></i></a>, <a class="txt" href="#E10:37_2_3_1_2_7"><i><font color="Green" title="E153">A147</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E13:37_2_3_1_2_7"/><b>then </b><i><font color="Green" title="E155">A149</font></i>: 
<font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">Fn</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E11:37_2_3_1_2"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="#E19:37_2_3_1_2"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="#E5:37_2_3_1_2_7"><i><font color="Green" title="E149">A143</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">p</font> = 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">p</font>
<b>by </b><i><a class="txt" href="#E102:37_2_3_1_2"><i><font color="Green" title="E127">A121</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_7"><i><font color="Green" title="E145">A139</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">p</font>
<b>by </b><i><a class="txt" href="#E12:37_2_3_1_2_7"><i><font color="Green" title="E154">A148</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font></span>)</span>
<b>by </b><i><a class="txt" href="#E7:37_2_3_1_2"><i><font color="Green" title="E68">A55</font></i></a>, <a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E11:37_2_3_1_2"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="#E34:37_2_3_1_2"><i><font color="Green" title="E86">A73</font></i></a>, <a class="txt" href="#E8:37_2_3_1_2_7"><i><font color="Green" title="E151">A145</font></i></a>, <a class="txt" href="#E12:37_2_3_1_2_7"><i><font color="Green" title="E154">A148</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c27">Uit</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font></span>)</span>
<b>by </b><i><a class="txt" href="#E77:37_2_3_1_2"><i><font color="Green" title="E110">A97</font></i></a>, <a class="txt" href="#E13:37_2_3_1_2_7"><i><font color="Green" title="E155">A149</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font></span>)</span>
<b>by </b><i><a class="txt" href="#E104:37_2_3_1_2"><i><font color="Green" title="E129">A123</font></i></a>, <a class="txt" href="#E4:37_2_3_1_2_7"><i><font color="Green" title="E148">A142</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><font color="Maroon" title="c52">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">p</font></span>)</span>
<b>by </b><i><a class="txt" href="#E77:37_2_3_1_2"><i><font color="Green" title="E110">A97</font></i></a>, <a class="txt" href="#E84:37_2_3_1_2"><i><font color="Green" title="E115">A105</font></i></a>, <a class="txt" href="#E95:37_2_3_1_2"><i><font color="Green" title="E120">A112</font></i></a>, <a class="txt" href="#E9:37_2_3_1_2_7"><i><font color="Green" title="E152">A146</font></i></a>, <a class="txt" href="#E13:37_2_3_1_2_7"><i><font color="Green" title="E155">A149</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>
<br/>.= 
<font color="Maroon" title="c48">Fni1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">p</font>
<b>by </b><i><a class="txt" href="#E96:37_2_3_1_2"><i><font color="Green" title="E121">A113</font></i></a>, <a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E102:37_2_3_1_2"><i><font color="Green" title="E127">A121</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_7"><i><font color="Green" title="E145">A139</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E146:37_2_3_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c45">S1</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">Fn2</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c48">Fni1</font> &amp; <font color="Olive" title="b1">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E97:37_2_3_1_2"><i><font color="Green" title="E122">A114</font></i></a>, <a class="txt" href="#E98:37_2_3_1_2"><i><font color="Green" title="E123">A115</font></i></a>, <a class="txt" href="#E122:37_2_3_1_2"><i><font color="Green" title="E137">A131</font></i></a>, <a class="txt" href="#E124:37_2_3_1_2"><i><font color="Green" title="E138">A132</font></i></a>, <a class="txt" href="#E125:37_2_3_1_2"><i><font color="Green" title="E139">A133</font></i></a>, <a class="txt" href="#E128:37_2_3_1_2"><i><font color="Green" title="E140">A134</font></i></a>, <a class="txt" href="#E130:37_2_3_1_2"><i><font color="Green" title="E141">A135</font></i></a>, <a class="txt" href="#E136:37_2_3_1_2"><i><font color="Green" title="E142">A136</font></i></a>, <a class="txt" href="#E144:37_2_3_1_2"><i><font color="Green" title="E144">A138</font></i></a>, <a class="ref" href="topalg_3.html#T20" title="TOPALG_3:th.20">TOPALG_3:20</a></i>;<br/><b>hence </b><a NAME="E147:37_2_3_1_2"/>
<font color="Maroon" title="c51">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c45">S1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font> &amp; <font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/><a NAME="E148:37_2_3_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c51">Fn1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green" title="E29">Lm12</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><a NAME="E149:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E145">A150</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c51">Fn1</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/><a NAME="E150:37_2_3_1_2"/><i><font color="Green" title="E146">A151</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c51">Fn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c45">S1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E108:37_2_3_1_2"><i><font color="Green" title="E130">A124</font></i></a>, <a class="txt" href="#E111:37_2_3_1_2"><i><font color="Green" title="E132">A126</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E151:37_2_3_1_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</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> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_3_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c62">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c62">a</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> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font></span>)</span> implies <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_3_1_2_8"/><i><font color="Green" title="E147">A152</font></i>: 
<font color="Maroon" title="c62">a</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> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_3_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_3_1_2_8_1"/>
( <font color="Maroon" title="c62">a</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="c48">Fni1</font> or  not <font color="Maroon" title="c62">a</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="c48">Fni1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_3_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:37_2_3_1_2_8_1_1"/><i><font color="Green" title="E148">A153</font></i>: 
<font color="Maroon" title="c62">a</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="c48">Fni1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span><br/><div class="add"><a NAME="E2:37_2_3_1_2_8_1_1"/><b>then </b><i><font color="Green" title="E149">A154</font></i>: 
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E101:37_2_3_1_2"><i><font color="Green" title="E126">A120</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E3:37_2_3_1_2_8_1_1"/><i><font color="Green" title="E150">A155</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E4:37_2_3_1_2_8_1_1"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E2:37_2_3_1_2_8_1_1"><i><font color="Green" title="E149">A154</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:37_2_3_1_2_8_1_1"/><b>then </b><i><font color="Green" title="E151">A156</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E82:37_2_3_1_2"><i><font color="Green" title="E113">A103</font></i></a></i>;<br/><a NAME="E6:37_2_3_1_2_8_1_1"/><b>then </b>
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E14:37_2_3_1_2"><i><font color="Green" title="E73">A60</font></i></a>, <a class="txt" href="#E15:37_2_3_1_2"><i><font color="Green" title="E74">A61</font></i></a>, <a class="txt" href="#E93:37_2_3_1_2"><i><font color="Green" title="E118">A110</font></i></a>, <a class="txt" href="#E96:37_2_3_1_2"><i><font color="Green" title="E121">A113</font></i></a>, <a class="txt" href="#E2:37_2_3_1_2_8_1_1"><i><font color="Green" title="E149">A154</font></i></a>, <a class="txt" href="#E3:37_2_3_1_2_8_1_1"><i><font color="Green" title="E150">A155</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E7:37_2_3_1_2_8_1_1"/><b>then </b><i><font color="Green" title="E152">A157</font></i>: 
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T182" title="RELAT_1:th.182">RELAT_1:182</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font> = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:37_2_3_1_2_8"><i><font color="Green" title="E147">A152</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c48">Fni1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:37_2_3_1_2_8_1_1"><i><font color="Green" title="E148">A153</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E101:37_2_3_1_2"><i><font color="Green" title="E126">A120</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_8_1_1"><i><font color="Green" title="E148">A153</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c43">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E101:37_2_3_1_2"><i><font color="Green" title="E126">A120</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_8_1_1"><i><font color="Green" title="E148">A153</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E91:37_2_3_1_2"><i><font color="Green" title="E117">A109</font></i></a>, <a class="txt" href="#E96:37_2_3_1_2"><i><font color="Green" title="E121">A113</font></i></a>, <a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E101:37_2_3_1_2"><i><font color="Green" title="E126">A120</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_8_1_1"><i><font color="Green" title="E148">A153</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E7:37_2_3_1_2_8_1_1"><i><font color="Green" title="E152">A157</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E91:37_2_3_1_2"><i><font color="Green" title="E117">A109</font></i></a>, <a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E101:37_2_3_1_2"><i><font color="Green" title="E126">A120</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_8_1_1"><i><font color="Green" title="E148">A153</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E14:37_2_3_1_2"><i><font color="Green" title="E73">A60</font></i></a>, <a class="txt" href="#E15:37_2_3_1_2"><i><font color="Green" title="E74">A61</font></i></a>, <a class="txt" href="#E94:37_2_3_1_2"><i><font color="Green" title="E119">A111</font></i></a>, <a class="txt" href="#E95:37_2_3_1_2"><i><font color="Green" title="E120">A112</font></i></a>, <a class="txt" href="#E5:37_2_3_1_2_8_1_1"><i><font color="Green" title="E151">A156</font></i></a>, <a class="ref" href="topalg_3.html#T2" title="TOPALG_3:th.2">TOPALG_3:2</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_3_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:37_2_3_1_2_8_1_2"/><i><font color="Green" title="E148">A158</font></i>: 
not <font color="Maroon" title="c62">a</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="c48">Fni1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span><br/><div class="add"><a NAME="E2:37_2_3_1_2_8_1_2"/><b>then </b><i><font color="Green" title="E149">A159</font></i>: 
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E101:37_2_3_1_2"><i><font color="Green" title="E126">A120</font></i></a>, <a class="txt" href="#E111:37_2_3_1_2"><i><font color="Green" title="E132">A126</font></i></a>, <a class="txt" href="#E149:37_2_3_1_2"><i><font color="Green" title="E145">A150</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_8"><i><font color="Green" title="E147">A152</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font> = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:37_2_3_1_2_8"><i><font color="Green" title="E147">A152</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:37_2_3_1_2_8_1_2"><i><font color="Green" title="E148">A158</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:37_2_3_1_2_8_1_2"><i><font color="Green" title="E149">A159</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">Fn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E11:37_2_3_1_2"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="#E19:37_2_3_1_2"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="#E109:37_2_3_1_2"><i><font color="Green" title="E131">A125</font></i></a>, <a class="txt" href="#E2:37_2_3_1_2_8_1_2"><i><font color="Green" title="E149">A159</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E7:37_2_3_1_2"><i><font color="Green" title="E68">A55</font></i></a>, <a class="txt" href="#E109:37_2_3_1_2"><i><font color="Green" title="E131">A125</font></i></a>, <a class="txt" href="#E2:37_2_3_1_2_8_1_2"><i><font color="Green" title="E149">A159</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E152:37_2_3_1_2"/>
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c45">S1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c51">Fn1</font>
 <b>by </b><i><a class="txt" href="#E149:37_2_3_1_2"><i><font color="Green" title="E145">A150</font></i></a>, <a class="txt" href="#E150:37_2_3_1_2"><i><font color="Green" title="E146">A151</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><a NAME="E153:37_2_3_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E14:37_2"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="#E2:37_2_3"><i><font color="Green" title="E62">A33</font></i></a></i>;<br/><a NAME="E154:37_2_3_1_2"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c45">S1</font>
 <b>by </b><i><a class="txt" href="#E114:37_2_3_1_2"><i><font color="Green" title="E133">A127</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E155:37_2_3_1_2"/><b>then </b><i><font color="Green" title="E147">A160</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c45">S1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E156:37_2_3_1_2"/><i><font color="Green" title="E148">A161</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c51">Fn1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><b>then </b><i><font color="Green" title="E149">A162</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c41">N1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c45">S1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E108:37_2_3_1_2"><i><font color="Green" title="E130">A124</font></i></a>, <a class="txt" href="#E111:37_2_3_1_2"><i><font color="Green" title="E132">A126</font></i></a>, <a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c45">S1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E155:37_2_3_1_2"><i><font color="Green" title="E147">A160</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><i><font color="Green" title="E150">A163</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">Ft</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c41">N1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> 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> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E6:37"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> 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> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><a NAME="E159:37_2_3_1_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_3_1_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c62">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c62">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_3_1_2_11"/><i><font color="Green" title="E151">A164</font></i>: 
<font color="Maroon" title="c62">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span><br/>

<a NAME="E2:37_2_3_1_2_11"/><b>then </b><i><font color="Green" title="E152">A165</font></i>: 
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E156:37_2_3_1_2"><i><font color="Green" title="E148">A161</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c63">a1</font>, <font color="Maroon" title="c64">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:37_2_3_1_2_11"/>
<font color="Maroon" title="c63">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>
 <b>and </b><br/><a NAME="E5:37_2_3_1_2_11"/><i><font color="Green" title="E153">A166</font></i>: 
<font color="Maroon" title="c64">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E6:37_2_3_1_2_11"/><i><font color="Green" title="E154">A167</font></i>: 
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c63">a1</font>,<font color="Maroon" title="c64">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E7:37_2_3_1_2_11"/><i><font color="Green" title="E155">A168</font></i>: 
<font color="Maroon" title="c64">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E5:37_2_3_1_2_11"><i><font color="Green" title="E153">A166</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:37_2_3_1_2_11"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">S</font>
 
<b>by </b><i><a class="txt" href="#E3:37_2_3_1_2"><i><font color="Green" title="E64">A51</font></i></a>, <a class="txt" href="#E20:37_2_3_1_2"><i><font color="Green" title="E78">A65</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E9:37_2_3_1_2_11"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">S</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:37_2_3_1_2_11"/><b>then </b><i><font color="Green" title="E156">A169</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E11:37_2_3_1_2_11"/><b>then </b><i><font color="Green" title="E157">A170</font></i>: 
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E157:37_2_3_1_2"><i><font color="Green" title="E149">A162</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_11"><i><font color="Green" title="E151">A164</font></i></a></i>;<br/>
<a NAME="E12:37_2_3_1_2_11"/><i><font color="Green" title="E158">A171</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E13:37_2_3_1_2_11"/><b>then </b><i><font color="Green" title="E159">A172</font></i>: 
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E11:37_2_3_1_2_11"><i><font color="Green" title="E157">A170</font></i></a></i>;<br/>
<a NAME="E14:37_2_3_1_2_11"/><i><font color="Green" title="E160">A173</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E76:37_2_3_1_2"><i><font color="Green" title="E109">A96</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_3_1_2_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_3_1_2_11_1"/>
(  not <font color="Maroon" title="c62">a</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="c48">Fni1</font> or <font color="Maroon" title="c62">a</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="c48">Fni1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_3_1_2_11_1_1"><b>suppose </b></a><a NAME="E1:37_2_3_1_2_11_1_1"/><i><font color="Green" title="E161">A174</font></i>: 
not <font color="Maroon" title="c62">a</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="c48">Fni1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span><br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font> = 
<font color="Maroon" title="c51">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E2:37_2_3_1_2_11"><i><font color="Green" title="E152">A165</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E1:37_2_3_1_2_11_1_1"><i><font color="Green" title="E161">A174</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E157:37_2_3_1_2"><i><font color="Green" title="E149">A162</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_11"><i><font color="Green" title="E151">A164</font></i></a>, <a class="txt" href="#E10:37_2_3_1_2_11"><i><font color="Green" title="E156">A169</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E8:37_2_3_1_2"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="#E2:37_2_3_1_2_11"><i><font color="Green" title="E152">A165</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c10">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E13:37_2_3_1_2_11"><i><font color="Green" title="E159">A172</font></i></a>, <a class="txt" href="#E14:37_2_3_1_2_11"><i><font color="Green" title="E160">A173</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E11:37_2_3_1_2_11"><i><font color="Green" title="E157">A170</font></i></a>, <a class="txt" href="#E12:37_2_3_1_2_11"><i><font color="Green" title="E158">A171</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_3_1_2_11_1_2"><b>suppose </b></a><a NAME="E1:37_2_3_1_2_11_1_2"/><i><font color="Green" title="E161">A175</font></i>: 
<font color="Maroon" title="c62">a</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="c48">Fni1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span><br/><div class="add"><a NAME="E2:37_2_3_1_2_11_1_2"/><b>then </b>
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E101:37_2_3_1_2"><i><font color="Green" title="E126">A120</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><b>then consider </b><font color="Maroon" title="c65">b1</font>, <font color="Maroon" title="c66">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:37_2_3_1_2_11_1_2"/><i><font color="Green" title="E162">A176</font></i>: 
<font color="Maroon" title="c65">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c41">N1</font>
 <b>and </b><br/><a NAME="E5:37_2_3_1_2_11_1_2"/><i><font color="Green" title="E163">A177</font></i>: 
<font color="Maroon" title="c66">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c44">SS</font>
 <b>and </b><br/><a NAME="E6:37_2_3_1_2_11_1_2"/><i><font color="Green" title="E164">A178</font></i>: 
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c65">b1</font>,<font color="Maroon" title="c66">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>set </b><font color="Maroon" title="c67">e</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>;<br/><i><font color="Green" title="E165">A179</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font> = 
<font color="Maroon" title="c10">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E11:37_2_3_1_2_11"><i><font color="Green" title="E157">A170</font></i></a>, <a class="txt" href="#E12:37_2_3_1_2_11"><i><font color="Green" title="E158">A171</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c22">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E13:37_2_3_1_2_11"><i><font color="Green" title="E159">A172</font></i></a>, <a class="txt" href="#E14:37_2_3_1_2_11"><i><font color="Green" title="E160">A173</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c21">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E8:37_2_3_1_2"><i><font color="Green" title="E69">A56</font></i></a>, <a class="txt" href="#E2:37_2_3_1_2_11"><i><font color="Green" title="E152">A165</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><a NAME="E8:37_2_3_1_2_11_1_2"/>
<font color="Maroon" title="c64">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c66">b2</font>
 <b>by </b><i><a class="txt" href="#E6:37_2_3_1_2_11"><i><font color="Green" title="E154">A167</font></i></a>, <a class="txt" href="#E6:37_2_3_1_2_11_1_2"><i><font color="Green" title="E164">A178</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:37_2_3_1_2_11_1_2"/><b>then </b><i><font color="Green" title="E166">A180</font></i>: 
<font color="Maroon" title="c64">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="#E20:37_2_3_1_2"><i><font color="Green" title="E78">A65</font></i></a>, <a class="txt" href="#E7:37_2_3_1_2_11"><i><font color="Green" title="E155">A168</font></i></a>, <a class="txt" href="#E5:37_2_3_1_2_11_1_2"><i><font color="Green" title="E163">A177</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/><a NAME="E10:37_2_3_1_2_11_1_2"/><i><font color="Green" title="E167">A181</font></i>: 
<font color="Maroon" title="c63">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c65">b1</font>
 <b>by </b><i><a class="txt" href="#E6:37_2_3_1_2_11"><i><font color="Green" title="E154">A167</font></i></a>, <a class="txt" href="#E6:37_2_3_1_2_11_1_2"><i><font color="Green" title="E164">A178</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E11:37_2_3_1_2_11_1_2"/><b>then </b><i><font color="Green" title="E168">A182</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c63">a1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c20">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E21:37_2_3_1_2"><i><font color="Green" title="E79">A66</font></i></a>, <a class="txt" href="#E4:37_2_3_1_2_11_1_2"><i><font color="Green" title="E162">A176</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:37_2_3_1_2_11_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c63">a1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E35:37_2_3_1_2"><i><font color="Green" title="E87">A74</font></i></a>, <a class="txt" href="#E4:37_2_3_1_2_11_1_2"><i><font color="Green" title="E162">A176</font></i></a>, <a class="txt" href="#E10:37_2_3_1_2_11_1_2"><i><font color="Green" title="E167">A181</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E13:37_2_3_1_2_11_1_2"/><b>then </b><i><font color="Green" title="E169">A183</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">Fn</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E10:37_2_3_1_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="txt" href="#E11:37_2_3_1_2"><i><font color="Green" title="E71">A58</font></i></a>, <a class="txt" href="#E19:37_2_3_1_2"><i><font color="Green" title="E77">A64</font></i></a>, <a class="txt" href="#E34:37_2_3_1_2"><i><font color="Green" title="E86">A73</font></i></a>, <a class="txt" href="#E109:37_2_3_1_2"><i><font color="Green" title="E131">A125</font></i></a>, <a class="txt" href="#E6:37_2_3_1_2_11"><i><font color="Green" title="E154">A167</font></i></a>, <a class="txt" href="#E7:37_2_3_1_2_11_1_2"><i><font color="Green" title="E165">A179</font></i></a>, <a class="txt" href="#E9:37_2_3_1_2_11_1_2"><i><font color="Green" title="E166">A180</font></i></a>, <a class="txt" href="#E11:37_2_3_1_2_11_1_2"><i><font color="Green" title="E168">A182</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>then </b><i><font color="Green" title="E170">A184</font></i>: <font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span> = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E77:37_2_3_1_2"><i><font color="Green" title="E110">A97</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E11:37_2_3_1_2_11"><i><font color="Green" title="E157">A170</font></i></a>, <a class="txt" href="#E12:37_2_3_1_2_11"><i><font color="Green" title="E158">A171</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">Ft</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E6:37"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="#E2:37_2_3_1_2_11"><i><font color="Green" title="E152">A165</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E3:37"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="#E2:37_2_3_1_2_11"><i><font color="Green" title="E152">A165</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font> = 
<font color="Maroon" title="c51">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E2:37_2_3_1_2_11"><i><font color="Green" title="E152">A165</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c48">Fni1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E1:37_2_3_1_2_11_1_2"><i><font color="Green" title="E161">A175</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,<font color="Maroon" title="c44">SS</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E96:37_2_3_1_2"><i><font color="Green" title="E121">A113</font></i></a>, <a class="txt" href="#E101:37_2_3_1_2"><i><font color="Green" title="E126">A120</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_11_1_2"><i><font color="Green" title="E161">A175</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c47">ff</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E99:37_2_3_1_2"><i><font color="Green" title="E124">A116</font></i></a>, <a class="txt" href="#E101:37_2_3_1_2"><i><font color="Green" title="E126">A120</font></i></a>, <a class="txt" href="#E1:37_2_3_1_2_11_1_2"><i><font color="Green" title="E161">A175</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c62">a</font>
<b>by </b><i><a class="txt" href="#E77:37_2_3_1_2"><i><font color="Green" title="E110">A97</font></i></a>, <a class="txt" href="#E84:37_2_3_1_2"><i><font color="Green" title="E115">A105</font></i></a>, <a class="txt" href="#E95:37_2_3_1_2"><i><font color="Green" title="E120">A112</font></i></a>, <a class="txt" href="#E13:37_2_3_1_2_11_1_2"><i><font color="Green" title="E169">A183</font></i></a>, <a class="txt" href="#E14:37_2_3_1_2_11_1_2"><i><font color="Green" title="E170">A184</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E160:37_2_3_1_2"/>
<font color="Maroon" title="c51">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c41">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E157:37_2_3_1_2"><i><font color="Green" title="E149">A162</font></i></a>, <a class="txt" href="#E158:37_2_3_1_2"><i><font color="Green" title="E150">A163</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E18:37_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E16:37_2"><i><font color="Green" title="E59">A30</font></i></a>, <a class="txt" href="#E17:37_2"><i><font color="Green" title="E60">A31</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c18">N2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c19">S</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> , <font color="Maroon" title="c20">Fn1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">S</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <b> such that </b><br/><a NAME="E20:37_2"/><i><font color="Green" title="E61">A185</font></i>: 
<font color="Maroon" title="c19">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">TT</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>and </b><br/><a NAME="E21:37_2"/><i><font color="Green" title="E62">A186</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">N2</font>
 <b>and </b><br/><a NAME="E22:37_2"/><i><font color="Green" title="E63">A187</font></i>: 
<font color="Maroon" title="c18">N2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font>
 <b>and </b><br/><a NAME="E23:37_2"/><i><font color="Green" title="E64">A188</font></i>: 
<font color="Maroon" title="c20">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E24:37_2"/><i><font color="Green" title="E65">A189</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c19">S</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">Fn1</font>
 <b>and </b><br/><a NAME="E25:37_2"/><i><font color="Green" title="E66">A190</font></i>: 
<font color="Maroon" title="c20">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E13:37_2"><i><font color="Green" title="E56">A20</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">z</font> = <font color="Maroon" title="c20">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c21">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c21">z</font>]</span><br/>

<b>take </b>
<font color="Maroon" title="c13">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ex <font color="Olive" title="b2">N</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> ex <font color="Olive" title="b3">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Olive" title="b1">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b3">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b2">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">Fn</font> &amp; <font color="Olive" title="b3">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b2">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; ( for <font color="Olive" title="b4">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b4">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b2">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">H</font> &amp; <font color="Olive" title="b4">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b2">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Olive" title="b3">Fn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">H</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c14">t</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">N</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> ex <font color="Olive" title="b2">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b2">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">Fn</font> &amp; <font color="Olive" title="b2">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; ( for <font color="Olive" title="b3">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b3">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b3">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Olive" title="b2">Fn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">H</font> ) )</span><br/>

<b>take </b>
<font color="Maroon" title="c18">N2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Fn</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">N2</font> &amp; <font color="Olive" title="b1">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">Fn</font> &amp; <font color="Olive" title="b1">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; ( for <font color="Olive" title="b2">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b2">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">H</font> &amp; <font color="Olive" title="b2">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Olive" title="b1">Fn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">H</font> ) )</span><br/>

<a NAME="E27:37_2"/><i><font color="Green" title="E67">A191</font></i>: 
<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c19">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="#E5:37_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="#E20:37_2"><i><font color="Green" title="E61">A185</font></i></a>, <a class="txt" href="topalg_5.html#E7"><i><font color="Green" title="E6">Lm6</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="tsep_1.html#T3" title="TSEP_1:th.3">TSEP_1:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">Fn1</font> = <font color="Maroon" title="c20">Fn1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c22">Fn1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">N2</font> &amp; <font color="Maroon" title="c22">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">Fn1</font> &amp; <font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; ( for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c22">Fn1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font> ) )</span><br/>

<b>thus </b><a NAME="E29:37_2"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">N2</font> &amp; <font color="Maroon" title="c22">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">Fn1</font> &amp; <font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="#E2:37_2"><i><font color="Green" title="E48">A12</font></i></a>, <a class="txt" href="#E5:37_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="#E20:37_2"><i><font color="Green" title="E61">A185</font></i></a>, <a class="txt" href="#E21:37_2"><i><font color="Green" title="E62">A186</font></i></a>, <a class="txt" href="#E23:37_2"><i><font color="Green" title="E64">A188</font></i></a>, <a class="txt" href="#E24:37_2"><i><font color="Green" title="E65">A189</font></i></a>, <a class="txt" href="#E25:37_2"><i><font color="Green" title="E66">A190</font></i></a>, <a class="txt" href="#E27:37_2"><i><font color="Green" title="E67">A191</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c22">Fn1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font></span><br/>

<b>let </b><font color="Maroon" title="c23">H</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font> &amp; <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <font color="Maroon" title="c22">Fn1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c23">H</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E30:37_2"/><i><font color="Green" title="E68">A192</font></i>: 
<font color="Maroon" title="c23">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E31:37_2"/><i><font color="Green" title="E69">A193</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font>
 <b>and </b><br/><a NAME="E32:37_2"/><i><font color="Green" title="E70">A194</font></i>: 
<font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">Fn1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c23">H</font></span><br/>

<a NAME="E33:37_2"/><i><font color="Green" title="E71">A195</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E34:37_2"/><i><font color="Green" title="E72">A196</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E35:37_2"/><i><font color="Green" title="E73">A197</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">N2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E36:37_2"/><i><font color="Green" title="E74">A198</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c22">Fn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> implies  ex <font color="Olive" title="b1">Z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> ) );<br/>
<a NAME="E37:37_2"/><i><font color="Green" title="E75">A199</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26" title="FINSEQ_3:th.26">FINSEQ_3:26</a></i>;<br/>
<a NAME="E38:37_2"/><i><font color="Green" title="E76">A200</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c24">i</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37_2_4"/><i><font color="Green" title="E77">A201</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c24">i</font>]
 <b>and </b><br/><a NAME="E2:37_2_4"/><i><font color="Green" title="E78">A202</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_4_1"/>
( <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font> )
 </span><b>by </b><i><a class="txt" href="#E2:37_2_4"><i><font color="Green" title="E78">A202</font></i></a>, <a class="ref" href="topreala.html#T23" title="TOPREALA:th.23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_4_1_1"><b>suppose </b></a><a NAME="E1:37_2_4_1_1"/><i><font color="Green" title="E79">A203</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )</span><br/><div class="add"><b>set </b><font color="Maroon" title="c25">Z</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/><a NAME="E2:37_2_4_1_1"/><i><font color="Green" title="E80">A204</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:37_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="#E1:37_2_4_1_1"><i><font color="Green" title="E79">A203</font></i></a>, <a class="ref" href="xxreal_1.html#T17" title="XXREAL_1:th.17">XXREAL_1:17</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">Z</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E4:37_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="#E1:37_2_4_1_1"><i><font color="Green" title="E79">A203</font></i></a>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="xxreal_1.html#T17" title="XXREAL_1:th.17">XXREAL_1:17</a></i>;<br/><b>take </b>
<font color="Maroon" title="c26">Z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )</span><br/><b>thus </b><a NAME="E4:37_2_4_1_1"/>
<font color="Maroon" title="c26">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span><br/><a NAME="E5:37_2_4_1_1"/><i><font color="Green" title="E81">A205</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/><a NAME="E6:37_2_4_1_1"/><b>then </b><i><font color="Green" title="E82">A206</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="txt" href="#E36:37_2"><i><font color="Green" title="E74">A198</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E7:37_2_4_1_1"/><i><font color="Green" title="E83">A207</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E33:37_2"><i><font color="Green" title="E71">A195</font></i></a>, <a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="txt" href="#E5:37_2_4_1_1"><i><font color="Green" title="E81">A205</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E8:37_2_4_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c27">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c27">x</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_4_1_1_1"/><i><font color="Green" title="E84">A208</font></i>: 
<font color="Maroon" title="c27">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c27">x</font></span><br/>

<a NAME="E2:37_2_4_1_1_1"/><i><font color="Green" title="E85">A209</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c26">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c27">x</font> = 
<font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c27">x</font>
<b>by </b><i><a class="txt" href="#E6:37_2_4_1_1"><i><font color="Green" title="E82">A206</font></i></a>, <a class="txt" href="#E1:37_2_4_1_1_1"><i><font color="Green" title="E84">A208</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c27">x</font>
<b>by </b><i><a class="txt" href="#E2:37_2_4_1_1"><i><font color="Green" title="E80">A204</font></i></a>, <a class="txt" href="#E6:37_2_4_1_1"><i><font color="Green" title="E82">A206</font></i></a>, <a class="txt" href="#E1:37_2_4_1_1_1"><i><font color="Green" title="E84">A208</font></i></a>, <a class="txt" href="#E2:37_2_4_1_1_1"><i><font color="Green" title="E85">A209</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c27">x</font>
<b>by </b><i><a class="txt" href="#E25:37_2"><i><font color="Green" title="E66">A190</font></i></a>, <a class="txt" href="#E32:37_2"><i><font color="Green" title="E70">A194</font></i></a>, <a class="txt" href="#E2:37_2_4_1_1"><i><font color="Green" title="E80">A204</font></i></a>, <a class="txt" href="#E6:37_2_4_1_1"><i><font color="Green" title="E82">A206</font></i></a>, <a class="txt" href="#E1:37_2_4_1_1_1"><i><font color="Green" title="E84">A208</font></i></a>, <a class="txt" href="#E2:37_2_4_1_1_1"><i><font color="Green" title="E85">A209</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c27">x</font>
<b>by </b><i><a class="txt" href="#E6:37_2_4_1_1"><i><font color="Green" title="E82">A206</font></i></a>, <a class="txt" href="#E1:37_2_4_1_1_1"><i><font color="Green" title="E84">A208</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:37_2_4_1_1"/>
<font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c26">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E6:37_2_4_1_1"><i><font color="Green" title="E82">A206</font></i></a>, <a class="txt" href="#E7:37_2_4_1_1"><i><font color="Green" title="E83">A207</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_4_1_2"><b>suppose </b></a><a NAME="E1:37_2_4_1_2"/><i><font color="Green" title="E79">A210</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c25">Z</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E3:37_2_4_1_2"/><i><font color="Green" title="E80">A211</font></i>: 
<font color="Maroon" title="c25">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>and </b><br/><a NAME="E4:37_2_4_1_2"/><i><font color="Green" title="E81">A212</font></i>: 
<font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E1:37_2_4"><i><font color="Green" title="E77">A201</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c26">ZZ</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/><a NAME="E5:37_2_4_1_2"/><i><font color="Green" title="E82">A213</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font>
 <b>by </b><i><a class="txt" href="#E14:37_2"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2"><i><font color="Green" title="E79">A210</font></i></a></i>;<br/><a NAME="E6:37_2_4_1_2"/><i><font color="Green" title="E83">A214</font></i>: 
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="#E14:37_2"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="#E2:37_2_4"><i><font color="Green" title="E78">A202</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2"><i><font color="Green" title="E79">A210</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">ZZ</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E15:37_2"><i><font color="Green" title="E58">A27</font></i></a>, <a class="txt" href="#E5:37_2_4_1_2"><i><font color="Green" title="E82">A213</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c28">Z1</font> = <font color="Maroon" title="c25">Z</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c27">ZZ</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c28">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )</span><br/><a NAME="E8:37_2_4_1_2"/><i><font color="Green" title="E84">A215</font></i>: 
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E14:37_2"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="#E2:37_2_4"><i><font color="Green" title="E78">A202</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2"><i><font color="Green" title="E79">A210</font></i></a></i>;<br/><b>hence </b><a NAME="E9:37_2_4_1_2"/>
<font color="Maroon" title="c28">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="#E3:37_2_4_1_2"><i><font color="Green" title="E80">A211</font></i></a>, <a class="txt" href="#E5:37_2_4_1_2"><i><font color="Green" title="E82">A213</font></i></a>, <a class="ref" href="xxreal_1.html#T165" title="XXREAL_1:th.165">XXREAL_1:165</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span><br/><a NAME="E10:37_2_4_1_2"/><i><font color="Green" title="E85">A216</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/><a NAME="E11:37_2_4_1_2"/><b>then </b><i><font color="Green" title="E86">A217</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="txt" href="#E36:37_2"><i><font color="Green" title="E74">A198</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E12:37_2_4_1_2"/><i><font color="Green" title="E87">A218</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E33:37_2"><i><font color="Green" title="E71">A195</font></i></a>, <a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="txt" href="#E10:37_2_4_1_2"><i><font color="Green" title="E85">A216</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E13:37_2_4_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_4_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c29">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c29">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_4_1_2_1"/><i><font color="Green" title="E88">A219</font></i>: 
<font color="Maroon" title="c29">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font></span><br/>

<b>consider </b><font color="Maroon" title="c30">x1</font>, <font color="Maroon" title="c31">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_2_4_1_2_1"/><i><font color="Green" title="E89">A220</font></i>: 
<font color="Maroon" title="c30">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">N2</font>
 <b>and </b><br/><a NAME="E4:37_2_4_1_2_1"/><i><font color="Green" title="E90">A221</font></i>: 
<font color="Maroon" title="c31">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">Z1</font>
 <b>and </b><br/><a NAME="E5:37_2_4_1_2_1"/><i><font color="Green" title="E91">A222</font></i>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<font color="Maroon" title="c31">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E11:37_2_4_1_2"><i><font color="Green" title="E86">A217</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1"><i><font color="Green" title="E88">A219</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">xy</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c30">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> <b>by </b><i><a class="txt" href="#E3:37_2_4_1_2_1"><i><font color="Green" title="E89">A220</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:37_2_4_1_2_1"/><i><font color="Green" title="E92">A223</font></i>: 
<font color="Maroon" title="c32">xy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">N2</font>
 
<b>by </b><i><a class="txt" href="#E3:37_2_4_1_2_1"><i><font color="Green" title="E89">A220</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c33">xZZ</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c34">Ui</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E10:37_2_4_1_2_1"/><i><font color="Green" title="E93">A224</font></i>: 
<font color="Maroon" title="c34">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">UL</font>
 <b>and </b><br/><a NAME="E11:37_2_4_1_2_1"/><i><font color="Green" title="E94">A225</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c34">Ui</font>
 <b>by </b><i><a class="txt" href="#E10:37_2"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E2:37_2_4"><i><font color="Green" title="E78">A202</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2"><i><font color="Green" title="E79">A210</font></i></a></i>;<br/>
<a NAME="E12:37_2_4_1_2_1"/><i><font color="Green" title="E95">A226</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c34">Ui</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">Ui</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c35">D</font> being  <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <b> such that </b><br/><a NAME="E14:37_2_4_1_2_1"/><i><font color="Green" title="E96">A227</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c35">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c34">Ui</font>
 <b>and </b><br/><a NAME="E15:37_2_4_1_2_1"/><i><font color="Green" title="E97">A228</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">D</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c34">Ui</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Olive" title="b2">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 <b>by </b><i><a class="txt" href="#E10:37"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="#E10:37_2_4_1_2_1"><i><font color="Green" title="E93">A224</font></i></a></i>;<br/>
<a NAME="E16:37_2_4_1_2_1"/>
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">Z</font>
 
<b>by </b><i><a class="txt" href="#E3:37_2_4_1_2"><i><font color="Green" title="E80">A211</font></i></a>, <a class="txt" href="#E5:37_2_4_1_2"><i><font color="Green" title="E82">A213</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E17:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E98">A229</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:37_2_4_1_2_1"><i><font color="Green" title="E89">A220</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E18:37_2_4_1_2_1"/><i><font color="Green" title="E99">A230</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<b>then </b><i><font color="Green" title="E100">A231</font></i>: <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E5:37_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="#E20:37_2"><i><font color="Green" title="E61">A185</font></i></a>, <a class="txt" href="#E24:37_2"><i><font color="Green" title="E65">A189</font></i></a>, <a class="txt" href="#E17:37_2_4_1_2_1"><i><font color="Green" title="E98">A229</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="txt" href="#E36:37_2"><i><font color="Green" title="E74">A198</font></i></a>, <a class="txt" href="#E17:37_2_4_1_2_1"><i><font color="Green" title="E98">A229</font></i></a>, <a class="txt" href="#E18:37_2_4_1_2_1"><i><font color="Green" title="E99">A230</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E20:37_2_4_1_2_1"/><i><font color="Green" title="E101">A232</font></i>: 
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">ZZ</font>
 
<b>by </b><i><a class="txt" href="#E8:37_2_4_1_2"><i><font color="Green" title="E84">A215</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E21:37_2_4_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E22:37_2"><i><font color="Green" title="E63">A187</font></i></a>, <a class="txt" href="#E3:37_2_4_1_2_1"><i><font color="Green" title="E89">A220</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E102">A233</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E23:37_2_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c34">Ui</font>
 
<b>by </b><i><a class="txt" href="#E11:37_2_4_1_2_1"><i><font color="Green" title="E94">A225</font></i></a>, <a class="txt" href="#E19:37_2_4_1_2_1"><i><font color="Green" title="E100">A231</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c36">Uit</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E25:37_2_4_1_2_1"/><i><font color="Green" title="E103">A234</font></i>: 
<font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Uit</font>
 <b>and </b><br/><a NAME="E26:37_2_4_1_2_1"/><i><font color="Green" title="E104">A235</font></i>: 
<font color="Maroon" title="c36">Uit</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">D</font>
 <b>by </b><i><a class="txt" href="#E14:37_2_4_1_2_1"><i><font color="Green" title="E96">A227</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c37">Uit</font> = <font color="Maroon" title="c36">Uit</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="#E25:37_2_4_1_2_1"><i><font color="Green" title="E103">A234</font></i></a>, <a class="txt" href="#E26:37_2_4_1_2_1"><i><font color="Green" title="E104">A235</font></i></a></i>;<br/>
<a NAME="E28:37_2_4_1_2_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font> &amp; <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E14:37_2"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="#E2:37_2_4"><i><font color="Green" title="E78">A202</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2"><i><font color="Green" title="E79">A210</font></i></a></i>;<br/>
<a NAME="E29:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E105">A236</font></i>: 
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<a NAME="E30:37_2_4_1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E14:37_2"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="#E2:37_2_4"><i><font color="Green" title="E78">A202</font></i></a></i>;<br/>
<a NAME="E31:37_2_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="#E6:37_2_4_1_2"><i><font color="Green" title="E83">A214</font></i></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<a NAME="E32:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E106">A237</font></i>: 
<font color="Maroon" title="c27">ZZ</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="#E8:37_2_4_1_2"><i><font color="Green" title="E84">A215</font></i></a>, <a class="txt" href="#E29:37_2_4_1_2_1"><i><font color="Green" title="E105">A236</font></i></a>, <a class="ref" href="borsuk_4.html#T49" title="BORSUK_4:th.49">BORSUK_4:49</a></i>;<br/>
<a NAME="E33:37_2_4_1_2_1"/>
<font color="Maroon" title="c32">xy</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="#E3:37_2_4_1_2_1"><i><font color="Green" title="E89">A220</font></i></a>, <a class="ref" href="connsp_1.html#T29" title="CONNSP_1:th.29">CONNSP_1:29</a></i>;<br/>
<a NAME="E34:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E107">A238</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="#E32:37_2_4_1_2_1"><i><font color="Green" title="E106">A237</font></i></a>, <a class="ref" href="topalg_3.html#T17" title="TOPALG_3:th.17">TOPALG_3:17</a></i>;<br/>
<a NAME="E35:37_2_4_1_2_1"/>
 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2" title="TSEP_1:th.2">TSEP_1:2</a></i>;<br/>
<a NAME="E36:37_2_4_1_2_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T25" title="BORSUK_3:th.25">BORSUK_3:25</a></i>;<br/>
<a NAME="E37:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E108">A239</font></i>: 
<font color="Maroon" title="c33">xZZ</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="#E34:37_2_4_1_2_1"><i><font color="Green" title="E107">A238</font></i></a>, <a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/>
<a NAME="E38:37_2_4_1_2_1"/><i><font color="Green" title="E109">A240</font></i>: 
<font color="Maroon" title="c30">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">xy</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E39:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E110">A241</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E20:37_2_4_1_2_1"><i><font color="Green" title="E101">A232</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E40:37_2_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E41:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E111">A242</font></i>: 
<font color="Maroon" title="c37">Uit</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c22">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E25:37_2_4_1_2_1"><i><font color="Green" title="E103">A234</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E42:37_2_4_1_2_1"/>
<font color="Maroon" title="c35">D</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c22">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_4_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c38">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font> or <font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c35">D</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_4_1_2_1_2"/>
<font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c35">D</font></span><br/>

<b>then consider </b><font color="Maroon" title="c39">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E3:37_2_4_1_2_1_2"/><i><font color="Green" title="E112">A243</font></i>: 
<font color="Maroon" title="c39">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">xZZ</font>
 <b>and </b><br/><a NAME="E4:37_2_4_1_2_1_2"/><i><font color="Green" title="E113">A244</font></i>: 
<font color="Maroon" title="c22">Fn1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green" title="E114">A245</font></i>: <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">Fn1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">a</font>
<b>by </b><i><a class="txt" href="#E5:37_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="#E20:37_2"><i><font color="Green" title="E61">A185</font></i></a>, <a class="txt" href="#E24:37_2"><i><font color="Green" title="E65">A189</font></i></a>, <a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:37_2_4_1_2_1_2"/>
<font color="Maroon" title="c32">xy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font>
 
<b>by </b><i><a class="txt" href="#E22:37_2"><i><font color="Green" title="E63">A187</font></i></a>, <a class="txt" href="#E3:37_2_4_1_2_1"><i><font color="Green" title="E89">A220</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:37_2_4_1_2_1_2"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E8:37_2_4_1_2_1_2"/><b>then </b><i><font color="Green" title="E115">A246</font></i>: 
<font color="Maroon" title="c39">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:37_2_4_1_2_1_2"><i><font color="Green" title="E112">A243</font></i></a></i>;<br/>
<a NAME="E9:37_2_4_1_2_1_2"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E10:37_2_4_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E8:37_2_4_1_2_1_2"><i><font color="Green" title="E115">A246</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E11:37_2_4_1_2_1_2"/>
<font color="Maroon" title="c38">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c35">D</font>
 <b>by </b><i><a class="txt" href="#E11:37_2_4_1_2_1"><i><font color="Green" title="E94">A225</font></i></a>, <a class="txt" href="#E14:37_2_4_1_2_1"><i><font color="Green" title="E96">A227</font></i></a>, <a class="txt" href="#E4:37_2_4_1_2_1_2"><i><font color="Green" title="E113">A244</font></i></a>, <a class="txt" href="#E5:37_2_4_1_2_1_2"><i><font color="Green" title="E114">A245</font></i></a>, <a class="txt" href="topalg_5.html#E35"><i><font color="Green" title="E29">Lm12</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E43:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E112">A247</font></i>: 
<font color="Maroon" title="c22">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c37">Uit</font>
 
<b>by </b><i><a class="txt" href="#E23:37_2"><i><font color="Green" title="E64">A188</font></i></a>, <a class="txt" href="#E27:37_2"><i><font color="Green" title="E67">A191</font></i></a>, <a class="txt" href="#E26:37_2_4_1_2_1"><i><font color="Green" title="E104">A235</font></i></a>, <a class="txt" href="#E37:37_2_4_1_2_1"><i><font color="Green" title="E108">A239</font></i></a>, <a class="txt" href="#E41:37_2_4_1_2_1"><i><font color="Green" title="E111">A242</font></i></a>, <a class="ref" href="topalg_3.html#T13" title="TOPALG_3:th.13">TOPALG_3:13</a>, <a class="ref" href="tops_2.html#T75" title="TOPS_2:th.75">TOPS_2:75</a></i>;<br/>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E31:37_2"><i><font color="Green" title="E69">A193</font></i></a>, <a class="txt" href="#E17:37_2_4_1_2_1"><i><font color="Green" title="E98">A229</font></i></a>, <a class="txt" href="#E18:37_2_4_1_2_1"><i><font color="Green" title="E99">A230</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E33:37_2"><i><font color="Green" title="E71">A195</font></i></a>, <a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="txt" href="#E17:37_2_4_1_2_1"><i><font color="Green" title="E98">A229</font></i></a>, <a class="txt" href="#E18:37_2_4_1_2_1"><i><font color="Green" title="E99">A230</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E45:37_2_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c34">Ui</font>
 
<b>by </b><i><a class="txt" href="#E11:37_2_4_1_2_1"><i><font color="Green" title="E94">A225</font></i></a>, <a class="txt" href="#E22:37_2_4_1_2_1"><i><font color="Green" title="E102">A233</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c38">Uith</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E47:37_2_4_1_2_1"/><i><font color="Green" title="E113">A248</font></i>: 
<font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c38">Uith</font>
 <b>and </b><br/><a NAME="E48:37_2_4_1_2_1"/><i><font color="Green" title="E114">A249</font></i>: 
<font color="Maroon" title="c38">Uith</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">D</font>
 <b>by </b><i><a class="txt" href="#E14:37_2_4_1_2_1"><i><font color="Green" title="E96">A227</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c39">Uith</font> = <font color="Maroon" title="c38">Uith</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="#E47:37_2_4_1_2_1"><i><font color="Green" title="E113">A248</font></i></a>, <a class="txt" href="#E48:37_2_4_1_2_1"><i><font color="Green" title="E114">A249</font></i></a></i>;<br/>
<a NAME="E50:37_2_4_1_2_1"/>
<font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E39:37_2_4_1_2_1"><i><font color="Green" title="E110">A241</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E51:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E115">A250</font></i>: 
<font color="Maroon" title="c39">Uith</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c23">H</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E47:37_2_4_1_2_1"><i><font color="Green" title="E113">A248</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E52:37_2_4_1_2_1"/>
<font color="Maroon" title="c35">D</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c23">H</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_4_1_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c40">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c40">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font> or <font color="Maroon" title="c40">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c35">D</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_4_1_2_1_4"/>
<font color="Maroon" title="c40">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c40">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c35">D</font></span><br/>

<b>then consider </b><font color="Maroon" title="c41">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c18">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E3:37_2_4_1_2_1_4"/><i><font color="Green" title="E116">A251</font></i>: 
<font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">xZZ</font>
 <b>and </b><br/><a NAME="E4:37_2_4_1_2_1_4"/><i><font color="Green" title="E117">A252</font></i>: 
<font color="Maroon" title="c23">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c41">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green" title="E118">A253</font></i>: <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c41">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c41">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">a</font>
<b>by </b><i><a class="txt" href="#E31:37_2"><i><font color="Green" title="E69">A193</font></i></a>, <a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:37_2_4_1_2_1_4"/>
<font color="Maroon" title="c32">xy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N</font>
 
<b>by </b><i><a class="txt" href="#E22:37_2"><i><font color="Green" title="E63">A187</font></i></a>, <a class="txt" href="#E3:37_2_4_1_2_1"><i><font color="Green" title="E89">A220</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:37_2_4_1_2_1_4"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E8:37_2_4_1_2_1_4"/><b>then </b><i><font color="Green" title="E119">A254</font></i>: 
<font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:37_2_4_1_2_1_4"><i><font color="Green" title="E116">A251</font></i></a></i>;<br/>
<a NAME="E9:37_2_4_1_2_1_4"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E10:37_2_4_1_2_1_4"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E8:37_2_4_1_2_1_4"><i><font color="Green" title="E119">A254</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E11:37_2_4_1_2_1_4"/>
<font color="Maroon" title="c40">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c35">D</font>
 <b>by </b><i><a class="txt" href="#E11:37_2_4_1_2_1"><i><font color="Green" title="E94">A225</font></i></a>, <a class="txt" href="#E14:37_2_4_1_2_1"><i><font color="Green" title="E96">A227</font></i></a>, <a class="txt" href="#E4:37_2_4_1_2_1_4"><i><font color="Green" title="E117">A252</font></i></a>, <a class="txt" href="#E5:37_2_4_1_2_1_4"><i><font color="Green" title="E118">A253</font></i></a>, <a class="txt" href="topalg_5.html#E35"><i><font color="Green" title="E29">Lm12</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E53:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E116">A255</font></i>: 
<font color="Maroon" title="c23">H</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c39">Uith</font>
 
<b>by </b><i><a class="txt" href="#E30:37_2"><i><font color="Green" title="E68">A192</font></i></a>, <a class="txt" href="#E37:37_2_4_1_2_1"><i><font color="Green" title="E108">A239</font></i></a>, <a class="txt" href="#E48:37_2_4_1_2_1"><i><font color="Green" title="E114">A249</font></i></a>, <a class="txt" href="#E51:37_2_4_1_2_1"><i><font color="Green" title="E115">A250</font></i></a>, <a class="ref" href="topalg_3.html#T13" title="TOPALG_3:th.13">TOPALG_3:13</a>, <a class="ref" href="tops_2.html#T75" title="TOPS_2:th.75">TOPS_2:75</a></i>;<br/>
<a NAME="E54:37_2_4_1_2_1"/><i><font color="Green" title="E117">A256</font></i>: 
<font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E33:37_2"><i><font color="Green" title="E71">A195</font></i></a>, <a class="txt" href="#E39:37_2_4_1_2_1"><i><font color="Green" title="E110">A241</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E17:37_2_4_1_2_1"><i><font color="Green" title="E98">A229</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E4:37_2_4_1_2"><i><font color="Green" title="E81">A212</font></i></a>, <a class="txt" href="#E17:37_2_4_1_2_1"><i><font color="Green" title="E98">A229</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E56:37_2_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c37">Uit</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c39">Uith</font>
 
<b>by </b><i><a class="txt" href="#E25:37_2_4_1_2_1"><i><font color="Green" title="E103">A234</font></i></a>, <a class="txt" href="#E53:37_2_4_1_2_1"><i><font color="Green" title="E116">A255</font></i></a>, <a class="txt" href="#E54:37_2_4_1_2_1"><i><font color="Green" title="E117">A256</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E57:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E118">A257</font></i>: 
<font color="Maroon" title="c37">Uit</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">Uith</font>
 
<b>by </b><i><a class="txt" href="#E26:37_2_4_1_2_1"><i><font color="Green" title="E104">A235</font></i></a>, <a class="txt" href="#E48:37_2_4_1_2_1"><i><font color="Green" title="E114">A249</font></i></a>, <a class="ref" href="taxonom2.html#D5" title="TAXONOM2:def.5">TAXONOM2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c40">f</font> = <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c37">Uit</font>;<br/>
<a NAME="E58:37_2_4_1_2_1"/><i><font color="Green" title="E119">A258</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c37">Uit</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">Uit</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E59:37_2_4_1_2_1"/><i><font color="Green" title="E120">A259</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c37">Uit</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">Uit</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green" title="E29">Lm12</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E60:37_2_4_1_2_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c37">Uit</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c34">Ui</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_4_1_2_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c41">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c41">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c37">Uit</font></span>)</span> or <font color="Maroon" title="c41">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Ui</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_4_1_2_1_6"/>
<font color="Maroon" title="c41">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c37">Uit</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c41">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Ui</font></span><br/>

<b>then consider </b><font color="Maroon" title="c42">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_2_4_1_2_1_6"/><i><font color="Green" title="E121">A260</font></i>: 
<font color="Maroon" title="c42">a</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> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c37">Uit</font></span>)</span>
 <b>and </b><br/><a NAME="E4:37_2_4_1_2_1_6"/><i><font color="Green" title="E122">A261</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c37">Uit</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:37_2_4_1_2_1_6"/>
<font color="Maroon" title="c42">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c35">D</font>
 
<b>by </b><i><a class="txt" href="#E26:37_2_4_1_2_1"><i><font color="Green" title="E104">A235</font></i></a>, <a class="txt" href="#E59:37_2_4_1_2_1"><i><font color="Green" title="E120">A259</font></i></a>, <a class="txt" href="#E3:37_2_4_1_2_1_6"><i><font color="Green" title="E121">A260</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:37_2_4_1_2_1_6"/><b>then </b>
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Ui</font>
 
<b>by </b><i><a class="txt" href="#E14:37_2_4_1_2_1"><i><font color="Green" title="E96">A227</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E7:37_2_4_1_2_1_6"/>
<font color="Maroon" title="c41">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Ui</font>
 <b>by </b><i><a class="txt" href="#E59:37_2_4_1_2_1"><i><font color="Green" title="E120">A259</font></i></a>, <a class="txt" href="#E3:37_2_4_1_2_1_6"><i><font color="Green" title="E121">A260</font></i></a>, <a class="txt" href="#E4:37_2_4_1_2_1_6"><i><font color="Green" title="E122">A261</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c41">f</font> = <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c37">Uit</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c37">Uit</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c34">Ui</font></span>)</span> <b>by </b><i><a class="txt" href="#E12:37_2_4_1_2_1"><i><font color="Green" title="E95">A226</font></i></a>, <a class="txt" href="#E58:37_2_4_1_2_1"><i><font color="Green" title="E119">A258</font></i></a>, <a class="txt" href="#E59:37_2_4_1_2_1"><i><font color="Green" title="E120">A259</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E62:37_2_4_1_2_1"/>
<font color="Maroon" title="c41">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 
<b>by </b><i><a class="txt" href="#E15:37_2_4_1_2_1"><i><font color="Green" title="E97">A228</font></i></a>, <a class="txt" href="#E26:37_2_4_1_2_1"><i><font color="Green" title="E104">A235</font></i></a></i>;<br/>
<a NAME="E63:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E121">A262</font></i>: 
<font color="Maroon" title="c41">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E64:37_2_4_1_2_1"/><i><font color="Green" title="E122">A263</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="txt" href="#E36:37_2"><i><font color="Green" title="E74">A198</font></i></a>, <a class="txt" href="#E7:37_2_4_1_2_1"><i><font color="Green" title="E92">A223</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E65:37_2_4_1_2_1"/><i><font color="Green" title="E123">A264</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E33:37_2"><i><font color="Green" title="E71">A195</font></i></a>, <a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="txt" href="#E7:37_2_4_1_2_1"><i><font color="Green" title="E92">A223</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E66:37_2_4_1_2_1"/><i><font color="Green" title="E124">A265</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c41">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_4_1_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c42">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> or <font color="Maroon" title="c42">b</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="c41">f</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_4_1_2_1_7"/>
<font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c42">b</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="c41">f</font></span><br/>

<b>then consider </b><font color="Maroon" title="c43">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_2_4_1_2_1_7"/><i><font color="Green" title="E125">A266</font></i>: 
<font color="Maroon" title="c43">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:37_2_4_1_2_1_7"/><i><font color="Green" title="E126">A267</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c43">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:37_2_4_1_2_1_7"/>
<font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c43">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">b</font>
 
<b>by </b><i><a class="txt" href="#E64:37_2_4_1_2_1"><i><font color="Green" title="E122">A263</font></i></a>, <a class="txt" href="#E3:37_2_4_1_2_1_7"><i><font color="Green" title="E125">A266</font></i></a>, <a class="txt" href="#E4:37_2_4_1_2_1_7"><i><font color="Green" title="E126">A267</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:37_2_4_1_2_1_7"/><b>then </b>
<font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E36:37_2"><i><font color="Green" title="E74">A198</font></i></a>, <a class="txt" href="#E64:37_2_4_1_2_1"><i><font color="Green" title="E122">A263</font></i></a>, <a class="txt" href="#E3:37_2_4_1_2_1_7"><i><font color="Green" title="E125">A266</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:37_2_4_1_2_1_7"/>
<font color="Maroon" title="c42">b</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="c41">f</font>
 <b>by </b><i><a class="txt" href="#E43:37_2_4_1_2_1"><i><font color="Green" title="E112">A247</font></i></a>, <a class="txt" href="#E59:37_2_4_1_2_1"><i><font color="Green" title="E120">A259</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E67:37_2_4_1_2_1"/><i><font color="Green" title="E125">A268</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c41">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_4_1_2_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c42">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> or <font color="Maroon" title="c42">b</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="c41">f</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_4_1_2_1_8"/>
<font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c42">b</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="c41">f</font></span><br/>

<b>then consider </b><font color="Maroon" title="c43">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_2_4_1_2_1_8"/><i><font color="Green" title="E126">A269</font></i>: 
<font color="Maroon" title="c43">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:37_2_4_1_2_1_8"/><i><font color="Green" title="E127">A270</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c43">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:37_2_4_1_2_1_8"/>
<font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c43">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">b</font>
 
<b>by </b><i><a class="txt" href="#E65:37_2_4_1_2_1"><i><font color="Green" title="E123">A264</font></i></a>, <a class="txt" href="#E3:37_2_4_1_2_1_8"><i><font color="Green" title="E126">A269</font></i></a>, <a class="txt" href="#E4:37_2_4_1_2_1_8"><i><font color="Green" title="E127">A270</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:37_2_4_1_2_1_8"/><b>then </b>
<font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c33">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E33:37_2"><i><font color="Green" title="E71">A195</font></i></a>, <a class="txt" href="#E65:37_2_4_1_2_1"><i><font color="Green" title="E123">A264</font></i></a>, <a class="txt" href="#E3:37_2_4_1_2_1_8"><i><font color="Green" title="E126">A269</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:37_2_4_1_2_1_8"/>
<font color="Maroon" title="c42">b</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="c41">f</font>
 <b>by </b><i><a class="txt" href="#E53:37_2_4_1_2_1"><i><font color="Green" title="E116">A255</font></i></a>, <a class="txt" href="#E57:37_2_4_1_2_1"><i><font color="Green" title="E118">A257</font></i></a>, <a class="txt" href="#E59:37_2_4_1_2_1"><i><font color="Green" title="E120">A259</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E68:37_2_4_1_2_1"/><i><font color="Green" title="E126">A271</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E66:37_2_4_1_2_1"><i><font color="Green" title="E124">A265</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E69:37_2_4_1_2_1"/><i><font color="Green" title="E127">A272</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E67:37_2_4_1_2_1"><i><font color="Green" title="E125">A268</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E70:37_2_4_1_2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_4_1_2_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c42">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c42">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_4_1_2_1_9"/><i><font color="Green" title="E128">A273</font></i>: 
<font color="Maroon" title="c42">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font></span><br/>

<a NAME="E2:37_2_4_1_2_1_9"/><i><font color="Green" title="E129">A274</font></i>: 
<font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E36:37_2"><i><font color="Green" title="E74">A198</font></i></a>, <a class="txt" href="#E64:37_2_4_1_2_1"><i><font color="Green" title="E122">A263</font></i></a>, <a class="txt" href="#E68:37_2_4_1_2_1"><i><font color="Green" title="E126">A271</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1_9"><i><font color="Green" title="E128">A273</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E3:37_2_4_1_2_1_9"/><i><font color="Green" title="E130">A275</font></i>: 
<font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">H</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E33:37_2"><i><font color="Green" title="E71">A195</font></i></a>, <a class="txt" href="#E64:37_2_4_1_2_1"><i><font color="Green" title="E122">A263</font></i></a>, <a class="txt" href="#E68:37_2_4_1_2_1"><i><font color="Green" title="E126">A271</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1_9"><i><font color="Green" title="E128">A273</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">Fn1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="ref" href="relat_1.html#T112" title="RELAT_1:th.112">RELAT_1:112</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="txt" href="#E64:37_2_4_1_2_1"><i><font color="Green" title="E122">A263</font></i></a>, <a class="txt" href="#E68:37_2_4_1_2_1"><i><font color="Green" title="E126">A271</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1_9"><i><font color="Green" title="E128">A273</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c41">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E36:37_2"><i><font color="Green" title="E74">A198</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1_9"><i><font color="Green" title="E128">A273</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E43:37_2_4_1_2_1"><i><font color="Green" title="E112">A247</font></i></a>, <a class="txt" href="#E2:37_2_4_1_2_1_9"><i><font color="Green" title="E129">A274</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="txt" href="#E36:37_2"><i><font color="Green" title="E74">A198</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1_9"><i><font color="Green" title="E128">A273</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:37_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="#E20:37_2"><i><font color="Green" title="E61">A185</font></i></a>, <a class="txt" href="#E24:37_2"><i><font color="Green" title="E65">A189</font></i></a>, <a class="txt" href="#E31:37_2"><i><font color="Green" title="E69">A193</font></i></a>, <a class="txt" href="#E33:37_2"><i><font color="Green" title="E71">A195</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1_9"><i><font color="Green" title="E128">A273</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c41">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E53:37_2_4_1_2_1"><i><font color="Green" title="E116">A255</font></i></a>, <a class="txt" href="#E57:37_2_4_1_2_1"><i><font color="Green" title="E118">A257</font></i></a>, <a class="txt" href="#E3:37_2_4_1_2_1_9"><i><font color="Green" title="E130">A275</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="txt" href="#E33:37_2"><i><font color="Green" title="E71">A195</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1_9"><i><font color="Green" title="E128">A273</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">H</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="txt" href="#E64:37_2_4_1_2_1"><i><font color="Green" title="E122">A263</font></i></a>, <a class="txt" href="#E68:37_2_4_1_2_1"><i><font color="Green" title="E126">A271</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1_9"><i><font color="Green" title="E128">A273</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="ref" href="relat_1.html#T112" title="RELAT_1:th.112">RELAT_1:112</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E71:37_2_4_1_2_1"/><b>then </b><i><font color="Green" title="E128">A276</font></i>: 
<font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E64:37_2_4_1_2_1"><i><font color="Green" title="E122">A263</font></i></a>, <a class="txt" href="#E65:37_2_4_1_2_1"><i><font color="Green" title="E123">A264</font></i></a>, <a class="txt" href="#E68:37_2_4_1_2_1"><i><font color="Green" title="E126">A271</font></i></a>, <a class="txt" href="#E69:37_2_4_1_2_1"><i><font color="Green" title="E127">A272</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_4_1_2_1_10"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_4_1_2_1_10"/>
( <font color="Maroon" title="c31">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">ZZ</font> or  not <font color="Maroon" title="c31">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">ZZ</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_4_1_2_1_10_1"><b>suppose </b></a><a NAME="E1:37_2_4_1_2_1_10_1"/>
<font color="Maroon" title="c31">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">ZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font></span><br/><div class="add"><a NAME="E2:37_2_4_1_2_1_10_1"/><b>then </b><i><font color="Green" title="E129">A277</font></i>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E5:37_2_4_1_2_1"><i><font color="Green" title="E91">A222</font></i></a>, <a class="txt" href="#E38:37_2_4_1_2_1"><i><font color="Green" title="E109">A240</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font> = 
<font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
<b>by </b><i><a class="txt" href="#E11:37_2_4_1_2"><i><font color="Green" title="E86">A217</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1"><i><font color="Green" title="E88">A219</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
<b>by </b><i><a class="txt" href="#E2:37_2_4_1_2_1_10_1"><i><font color="Green" title="E129">A277</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c32">xy</font>,<font color="Maroon" title="c27">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
<b>by </b><i><a class="txt" href="#E63:37_2_4_1_2_1"><i><font color="Green" title="E121">A262</font></i></a>, <a class="txt" href="#E64:37_2_4_1_2_1"><i><font color="Green" title="E122">A263</font></i></a>, <a class="txt" href="#E65:37_2_4_1_2_1"><i><font color="Green" title="E123">A264</font></i></a>, <a class="txt" href="#E66:37_2_4_1_2_1"><i><font color="Green" title="E124">A265</font></i></a>, <a class="txt" href="#E67:37_2_4_1_2_1"><i><font color="Green" title="E125">A268</font></i></a>, <a class="txt" href="#E71:37_2_4_1_2_1"><i><font color="Green" title="E128">A276</font></i></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></i>
<br/>.= 
<font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
<b>by </b><i><a class="txt" href="#E2:37_2_4_1_2_1_10_1"><i><font color="Green" title="E129">A277</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
<b>by </b><i><a class="txt" href="#E11:37_2_4_1_2"><i><font color="Green" title="E86">A217</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1"><i><font color="Green" title="E88">A219</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_4_1_2_1_10_2"><b>suppose </b></a><a NAME="E1:37_2_4_1_2_1_10_2"/>
not <font color="Maroon" title="c31">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">ZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font></span><br/><div class="add"><a NAME="E2:37_2_4_1_2_1_10_2"/><b>then </b>
<font color="Maroon" title="c31">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">Z</font>
 <b>by </b><i><a class="txt" href="#E4:37_2_4_1_2_1"><i><font color="Green" title="E90">A221</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:37_2_4_1_2_1_10_2"/><b>then </b><i><font color="Green" title="E129">A278</font></i>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E3:37_2_4_1_2_1"><i><font color="Green" title="E89">A220</font></i></a>, <a class="txt" href="#E5:37_2_4_1_2_1"><i><font color="Green" title="E91">A222</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font> = 
<font color="Maroon" title="c22">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
<b>by </b><i><a class="txt" href="#E11:37_2_4_1_2"><i><font color="Green" title="E86">A217</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1"><i><font color="Green" title="E88">A219</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
<b>by </b><i><a class="txt" href="#E3:37_2_4_1_2_1_10_2"><i><font color="Green" title="E129">A278</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c23">H</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
<b>by </b><i><a class="txt" href="#E4:37_2_4_1_2"><i><font color="Green" title="E81">A212</font></i></a>, <a class="txt" href="#E3:37_2_4_1_2_1_10_2"><i><font color="Green" title="E129">A278</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
<b>by </b><i><a class="txt" href="#E11:37_2_4_1_2"><i><font color="Green" title="E86">A217</font></i></a>, <a class="txt" href="#E1:37_2_4_1_2_1"><i><font color="Green" title="E88">A219</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E14:37_2_4_1_2"/>
<font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c28">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E11:37_2_4_1_2"><i><font color="Green" title="E86">A217</font></i></a>, <a class="txt" href="#E12:37_2_4_1_2"><i><font color="Green" title="E87">A218</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E39:37_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E37:37_2"><i><font color="Green" title="E75">A199</font></i></a>, <a class="txt" href="#E38:37_2"><i><font color="Green" title="E76">A200</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c24">Z</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E41:37_2"/><i><font color="Green" title="E77">A279</font></i>: 
<font color="Maroon" title="c24">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c15">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">TT</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>and </b><br/><a NAME="E42:37_2"/><i><font color="Green" title="E78">A280</font></i>: 
<font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E13:37_2"><i><font color="Green" title="E56">A20</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c22">Fn1</font> = 
<font color="Maroon" title="c22">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c18">N2</font>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
<b>by </b><i><a class="txt" href="#E5:37_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="txt" href="#E36:37_2"><i><font color="Green" title="E74">A198</font></i></a>, <a class="txt" href="#E41:37_2"><i><font color="Green" title="E77">A279</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>
<br/>.= 
<font color="Maroon" title="c23">H</font>
<b>by </b><i><a class="txt" href="#E5:37_2"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="#E33:37_2"><i><font color="Green" title="E71">A195</font></i></a>, <a class="txt" href="#E34:37_2"><i><font color="Green" title="E72">A196</font></i></a>, <a class="txt" href="#E35:37_2"><i><font color="Green" title="E73">A197</font></i></a>, <a class="txt" href="#E41:37_2"><i><font color="Green" title="E77">A279</font></i></a>, <a class="txt" href="#E42:37_2"><i><font color="Green" title="E78">A280</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c12">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <b> such that </b><br/><a NAME="E14:37"/><i><font color="Green" title="E48">A281</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E12:37"><i><font color="Green" title="E47">A11</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c12">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font> &amp; <font color="Maroon" title="c12">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> &amp; ( for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> holds <br/><font color="Maroon" title="c12">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font> ) )</span><br/>

<a NAME="E15:37"/><i><font color="Green" title="E49">A282</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E4:37"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:37"/><i><font color="Green" title="E50">A283</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<b>consider </b><font color="Maroon" title="c14">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c15">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> , <font color="Maroon" title="c16">N</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c17">Fn</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <b> such that </b><br/><a NAME="E2:37_3"/><i><font color="Green" title="E51">A284</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E3:37_3"/><i><font color="Green" title="E52">A285</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>and </b><br/><a NAME="E4:37_3"/><i><font color="Green" title="E53">A286</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N</font>
 <b>and </b><br/><a NAME="E5:37_3"/>
<font color="Maroon" title="c17">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E6:37_3"/><i><font color="Green" title="E54">A287</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">Fn</font>
 <b>and </b><br/><a NAME="E7:37_3"/>
<font color="Maroon" title="c17">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E8:37_3"/>
for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c17">Fn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font>
 <b>by </b><i><a class="txt" href="#E14:37"><i><font color="Green" title="E48">A281</font></i></a></i>;<br/>
<i><font color="Green" title="E55">A288</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N</font></span>)</span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E10:37_3"/><b>then </b><i><font color="Green" title="E56">A289</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E2:37_3"><i><font color="Green" title="E51">A284</font></i></a>, <a class="txt" href="#E4:37_3"><i><font color="Green" title="E53">A286</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">Fn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="#E3:37_3"><i><font color="Green" title="E52">A285</font></i></a>, <a class="txt" href="#E10:37_3"><i><font color="Green" title="E56">A289</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="#E6:37_3"><i><font color="Green" title="E54">A287</font></i></a>, <a class="txt" href="#E9:37_3"><i><font color="Green" title="E55">A288</font></i></a>, <a class="txt" href="#E10:37_3"><i><font color="Green" title="E56">A289</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E51">A290</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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 <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  holds   <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c13">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><b>let </b><font color="Maroon" title="c14">F</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c13">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><b>thus </b> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">F</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">N</font></span>)</span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c13">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E18:37"/><i><font color="Green" title="E52">A291</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><br/> for <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c8">Y</font><br/> for <font color="Olive" title="b3">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b4">N1</font>, <font color="Olive" title="b5">N2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font><br/> for <font color="Olive" title="b6">Fn1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b4">N1</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <br/> for <font color="Olive" title="b7">Fn2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b5">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">N1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">N2</font> &amp; <font color="Olive" title="b7">Fn2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b6">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b5">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b7">Fn2</font> &amp; <font color="Olive" title="b7">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b5">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b4">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b6">Fn1</font> &amp; <font color="Olive" title="b6">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b4">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Olive" title="b6">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c8">Y</font><br/> for <font color="Olive" title="b2">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b3">N1</font>, <font color="Olive" title="b4">N2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font><br/> for <font color="Olive" title="b5">Fn1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">N1</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <br/> for <font color="Olive" title="b6">Fn2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b4">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">N1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">N2</font> &amp; <font color="Olive" title="b6">Fn2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b5">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b4">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b6">Fn2</font> &amp; <font color="Olive" title="b6">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b4">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b3">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">Fn1</font> &amp; <font color="Olive" title="b5">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b3">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Olive" title="b5">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><b>let </b><font color="Maroon" title="c14">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c8">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b2">N1</font>, <font color="Olive" title="b3">N2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font><br/> for <font color="Olive" title="b4">Fn1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">N1</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <br/> for <font color="Olive" title="b5">Fn2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Olive" title="b1">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N1</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">N2</font> &amp; <font color="Olive" title="b5">Fn2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b4">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b3">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">Fn2</font> &amp; <font color="Olive" title="b5">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b3">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b2">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">Fn1</font> &amp; <font color="Olive" title="b4">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b2">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Olive" title="b4">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><b>let </b><font color="Maroon" title="c15">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">N1</font>, <font color="Olive" title="b2">N2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font><br/> for <font color="Olive" title="b3">Fn1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">N1</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <br/> for <font color="Olive" title="b4">Fn2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N1</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N2</font> &amp; <font color="Olive" title="b4">Fn2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b3">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b2">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">Fn2</font> &amp; <font color="Olive" title="b4">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b2">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">Fn1</font> &amp; <font color="Olive" title="b3">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Olive" title="b3">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><b>let </b><font color="Maroon" title="c16">N1</font>, <font color="Maroon" title="c17">N2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Fn1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N1</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <br/> for <font color="Olive" title="b2">Fn2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N1</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N2</font> &amp; <font color="Olive" title="b2">Fn2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">Fn2</font> &amp; <font color="Olive" title="b2">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">Fn1</font> &amp; <font color="Olive" title="b1">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Olive" title="b1">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><b>let </b><font color="Maroon" title="c18">Fn1</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N1</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Fn2</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N1</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N2</font> &amp; <font color="Olive" title="b1">Fn2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c18">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">Fn2</font> &amp; <font color="Olive" title="b1">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Fn1</font> &amp; <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/><b>let </b><font color="Maroon" title="c19">Fn2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N1</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N2</font> &amp; <font color="Maroon" title="c19">Fn2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c18">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">Fn2</font> &amp; <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Fn1</font> &amp; <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>











<b>assume </b><b>that </b><br/><a NAME="E1:37_5"/>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E2:37_5"/><i><font color="Green" title="E53">A292</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N1</font>
 <b>and </b><br/><a NAME="E3:37_5"/><i><font color="Green" title="E54">A293</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N2</font>
 <b>and </b><br/><a NAME="E4:37_5"/><i><font color="Green" title="E55">A294</font></i>: 
<font color="Maroon" title="c19">Fn2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E5:37_5"/><i><font color="Green" title="E56">A295</font></i>: 
<font color="Maroon" title="c18">Fn1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E6:37_5"/><i><font color="Green" title="E57">A296</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">Fn2</font>
 <b>and </b><br/><a NAME="E7:37_5"/><i><font color="Green" title="E58">A297</font></i>: 
<font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E8:37_5"/><i><font color="Green" title="E59">A298</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Fn1</font>
 <b>and </b><br/><a NAME="E9:37_5"/><i><font color="Green" title="E60">A299</font></i>: 
<font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<a NAME="E10:37_5"/><i><font color="Green" title="E61">A300</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">N1</font>
 
<b>by </b><i><a class="txt" href="#E2:37_5"><i><font color="Green" title="E53">A292</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:37_5"/><i><font color="Green" title="E62">A301</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">Fn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E17:37"><i><font color="Green" title="E51">A290</font></i></a></i>;<br/>
<a NAME="E12:37_5"/><i><font color="Green" title="E63">A302</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N2</font>
 
<b>by </b><i><a class="txt" href="#E3:37_5"><i><font color="Green" title="E54">A293</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E13:37_5"/><i><font color="Green" title="E64">A303</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">Fn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E17:37"><i><font color="Green" title="E51">A290</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">TT</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E15:37_5"/><i><font color="Green" title="E65">A304</font></i>: 
<font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E16:37_5"/><i><font color="Green" title="E66">A305</font></i>: 
<font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">TT</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E17:37_5"/><i><font color="Green" title="E67">A306</font></i>: 
<font color="Maroon" title="c20">TT</font> is <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>
 <b>and </b><br/><a NAME="E18:37_5"/><i><font color="Green" title="E68">A307</font></i>: 
 ex <font color="Olive" title="b1">N</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> st <br/>( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">N</font> &amp; ( for <font color="Olive" title="b2">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font> &amp; <font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font> holds <br/> ex <font color="Olive" title="b3">Ui</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive" title="b3">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">UL</font> &amp; <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Ui</font> ) ) )
 <b>by </b><i><a class="txt" href="#E1:37"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="#E8:37"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E9:37"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="topalg_5.html#T21" target="_self" title="TOPALG_5:th.21">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">N</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font><b> such that </b><br/><a NAME="E20:37_5"/><i><font color="Green" title="E69">A308</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">N</font>
 <b>and </b><br/><a NAME="E21:37_5"/><i><font color="Green" title="E70">A309</font></i>: 
for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font> holds <br/> ex <font color="Olive" title="b2">Ui</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive" title="b2">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">UL</font> &amp; <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c21">N</font>,<span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Ui</font> )
 <b>by </b><i><a class="txt" href="#E18:37_5"><i><font color="Green" title="E68">A307</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">N</font> = <font color="Maroon" title="c21">N</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> <b>by </b><i><a class="txt" href="#E20:37_5"><i><font color="Green" title="E69">A308</font></i></a></i>;<br/>
<a NAME="E23:37_5"/><i><font color="Green" title="E71">A310</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E24:37_5"/><i><font color="Green" title="E72">A311</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">TT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E25:37_5"/><i><font color="Green" title="E73">A312</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">Fn2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E26:37_5"/><i><font color="Green" title="E74">A313</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N2</font></span>)</span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E27:37_5"/><i><font color="Green" title="E75">A314</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">N2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E28:37_5"/><i><font color="Green" title="E76">A315</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N1</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N1</font></span>)</span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E29:37_5"/><i><font color="Green" title="E77">A316</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">N1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E30:37_5"/><i><font color="Green" title="E78">A317</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">Fn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N1</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font> implies  ex <font color="Olive" title="b1">Z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> ) );<br/>
<a NAME="E31:37_5"/><i><font color="Green" title="E79">A318</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26" title="FINSEQ_3:th.26">FINSEQ_3:26</a></i>;<br/>
<a NAME="E32:37_5"/><i><font color="Green" title="E80">A319</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">i</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37_5_1"/><i><font color="Green" title="E81">A320</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c23">i</font>]
 <b>and </b><br/><a NAME="E2:37_5_1"/><i><font color="Green" title="E82">A321</font></i>: 
<font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_1_1"/>
( <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font> )
 </span><b>by </b><i><a class="txt" href="#E2:37_5_1"><i><font color="Green" title="E82">A321</font></i></a>, <a class="ref" href="topreala.html#T23" title="TOPREALA:th.23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_1"><b>suppose </b></a><a NAME="E1:37_5_1_1_1"/><i><font color="Green" title="E83">A322</font></i>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )</span><br/><div class="add"><b>set </b><font color="Maroon" title="c24">Z</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/><a NAME="E2:37_5_1_1_1"/><i><font color="Green" title="E84">A323</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E15:37_5"><i><font color="Green" title="E65">A304</font></i></a>, <a class="txt" href="#E1:37_5_1_1_1"><i><font color="Green" title="E83">A322</font></i></a>, <a class="ref" href="xxreal_1.html#T17" title="XXREAL_1:th.17">XXREAL_1:17</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">Z</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E15:37_5"><i><font color="Green" title="E65">A304</font></i></a>, <a class="txt" href="#E1:37_5_1_1_1"><i><font color="Green" title="E83">A322</font></i></a>, <a class="txt" href="topalg_5.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="xxreal_1.html#T17" title="XXREAL_1:th.17">XXREAL_1:17</a></i>;<br/><b>take </b>
<font color="Maroon" title="c25">Z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )</span><br/><b>thus </b><a NAME="E4:37_5_1_1_1"/>
<font color="Maroon" title="c25">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span><br/><a NAME="E5:37_5_1_1_1"/><i><font color="Green" title="E85">A324</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E10:37_5"><i><font color="Green" title="E61">A300</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E6:37_5_1_1_1"/><i><font color="Green" title="E86">A325</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E10:37_5"><i><font color="Green" title="E61">A300</font></i></a>, <a class="txt" href="#E11:37_5"><i><font color="Green" title="E62">A301</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E7:37_5_1_1_1"/><i><font color="Green" title="E87">A326</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E12:37_5"><i><font color="Green" title="E63">A302</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E8:37_5_1_1_1"/><i><font color="Green" title="E88">A327</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E12:37_5"><i><font color="Green" title="E63">A302</font></i></a>, <a class="txt" href="#E13:37_5"><i><font color="Green" title="E64">A303</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E9:37_5_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font> )</span><br/>

<b>assume </b><a NAME="E1:37_5_1_1_1_1"/><i><font color="Green" title="E89">A328</font></i>: 
<font color="Maroon" title="c26">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font></span><br/>

<a NAME="E2:37_5_1_1_1_1"/><i><font color="Green" title="E90">A329</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<font color="Maroon" title="c25">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font> = 
<font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="#E6:37_5_1_1_1"><i><font color="Green" title="E86">A325</font></i></a>, <a class="txt" href="#E1:37_5_1_1_1_1"><i><font color="Green" title="E89">A328</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="#E2:37_5_1_1_1"><i><font color="Green" title="E84">A323</font></i></a>, <a class="txt" href="#E6:37_5_1_1_1"><i><font color="Green" title="E86">A325</font></i></a>, <a class="txt" href="#E1:37_5_1_1_1_1"><i><font color="Green" title="E89">A328</font></i></a>, <a class="txt" href="#E2:37_5_1_1_1_1"><i><font color="Green" title="E90">A329</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c10">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="#E9:37_5"><i><font color="Green" title="E60">A299</font></i></a>, <a class="txt" href="#E5:37_5_1_1_1"><i><font color="Green" title="E85">A324</font></i></a>, <a class="txt" href="#E6:37_5_1_1_1"><i><font color="Green" title="E86">A325</font></i></a>, <a class="txt" href="#E1:37_5_1_1_1_1"><i><font color="Green" title="E89">A328</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="#E6:37_5_1_1_1"><i><font color="Green" title="E86">A325</font></i></a>, <a class="txt" href="#E7:37_5_1_1_1"><i><font color="Green" title="E87">A326</font></i></a>, <a class="txt" href="#E1:37_5_1_1_1_1"><i><font color="Green" title="E89">A328</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="#E7:37_5"><i><font color="Green" title="E58">A297</font></i></a>, <a class="txt" href="#E2:37_5_1_1_1"><i><font color="Green" title="E84">A323</font></i></a>, <a class="txt" href="#E6:37_5_1_1_1"><i><font color="Green" title="E86">A325</font></i></a>, <a class="txt" href="#E1:37_5_1_1_1_1"><i><font color="Green" title="E89">A328</font></i></a>, <a class="txt" href="#E2:37_5_1_1_1_1"><i><font color="Green" title="E90">A329</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
<b>by </b><i><a class="txt" href="#E6:37_5_1_1_1"><i><font color="Green" title="E86">A325</font></i></a>, <a class="txt" href="#E1:37_5_1_1_1_1"><i><font color="Green" title="E89">A328</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:37_5_1_1_1"/>
<font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c25">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E6:37_5_1_1_1"><i><font color="Green" title="E86">A325</font></i></a>, <a class="txt" href="#E8:37_5_1_1_1"><i><font color="Green" title="E88">A327</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_2"><b>suppose </b></a><a NAME="E1:37_5_1_1_2"/><i><font color="Green" title="E83">A330</font></i>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c24">Z</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E3:37_5_1_1_2"/><i><font color="Green" title="E84">A331</font></i>: 
<font color="Maroon" title="c24">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>and </b><br/><a NAME="E4:37_5_1_1_2"/><i><font color="Green" title="E85">A332</font></i>: 
<font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E1:37_5_1"><i><font color="Green" title="E81">A320</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c25">ZZ</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>;<br/><div><i><font color="Green" title="E86">A333</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font> implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> &amp; ( <font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font> implies ( <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) ) ) )</span><br/><b>assume </b><a NAME="E1:37_5_1_1_2_1"/><i><font color="Green" title="E87">A334</font></i>: 
<font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> &amp; ( <font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font> implies ( <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) ) )</span><br/><a NAME="E2:37_5_1_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">i</font>
 <b>by </b><i><a class="txt" href="#E1:37_5_1_1_2_1"><i><font color="Green" title="E87">A334</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:37_5_1_1_2_1"/><b>then </b>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">i</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">i</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E4:37_5_1_1_2_1"/><i><font color="Green" title="E88">A335</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 <b>by </b><i><a class="txt" href="#E15:37_5"><i><font color="Green" title="E65">A304</font></i></a>, <a class="txt" href="#E17:37_5"><i><font color="Green" title="E67">A306</font></i></a>, <a class="txt" href="#E23:37_5"><i><font color="Green" title="E71">A310</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_1"><i><font color="Green" title="E87">A334</font></i></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font> implies ( <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )</span><br/><b>assume </b><a NAME="E5:37_5_1_1_2_1"/><i><font color="Green" title="E89">A336</font></i>: 
<font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">TT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><a NAME="E6:37_5_1_1_2_1"/><i><font color="Green" title="E90">A337</font></i>: 
<font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E7:37_5_1_1_2_1"/><i><font color="Green" title="E91">A338</font></i>: 
<font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E17:37_5"><i><font color="Green" title="E67">A306</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_1"><i><font color="Green" title="E87">A334</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2_1"><i><font color="Green" title="E89">A336</font></i></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><a NAME="E8:37_5_1_1_2_1"/>
<font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">TT</font>
 <b>by </b><i><a class="txt" href="#E5:37_5_1_1_2_1"><i><font color="Green" title="E89">A336</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:37_5_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">TT</font> or <font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">TT</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E10:37_5_1_1_2_1"/>
<font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="#E16:37_5"><i><font color="Green" title="E66">A305</font></i></a>, <a class="txt" href="#E17:37_5"><i><font color="Green" title="E67">A306</font></i></a>, <a class="txt" href="#E24:37_5"><i><font color="Green" title="E72">A311</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2_1"><i><font color="Green" title="E89">A336</font></i></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><b>hence </b><a NAME="E11:37_5_1_1_2_1"/>
<font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="#E7:37_5_1_1_2_1"><i><font color="Green" title="E91">A338</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><b>thus </b><a NAME="E12:37_5_1_1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E17:37_5"><i><font color="Green" title="E67">A306</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_1"><i><font color="Green" title="E87">A334</font></i></a>, <a class="txt" href="#E4:37_5_1_1_2_1"><i><font color="Green" title="E88">A335</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2_1"><i><font color="Green" title="E89">A336</font></i></a>, <a class="txt" href="#E6:37_5_1_1_2_1"><i><font color="Green" title="E90">A337</font></i></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E87">A339</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 implies <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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>  )</span><br/><b>assume </b><a NAME="E1:37_5_1_1_2_2"/><i><font color="Green" title="E88">A340</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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> </span><br/><b>thus </b><a NAME="E2:37_5_1_1_2_2"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> or <font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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>  )</span><br/>

<b>assume </b><a NAME="E1:37_5_1_1_2_2_1"/><i><font color="Green" title="E89">A341</font></i>: 
<font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> </span><br/>

<b>then reconsider </b><font color="Maroon" title="c28">a</font> = <font color="Maroon" title="c27">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E3:37_5_1_1_2_2_1"/>
( <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">a</font> &amp; <font color="Maroon" title="c28">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:37_5_1_1_2_2_1"><i><font color="Green" title="E89">A341</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E4:37_5_1_1_2_2_1"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">a</font> &amp; <font color="Maroon" title="c28">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E1:37_5_1_1_2_2"><i><font color="Green" title="E88">A340</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:37_5_1_1_2_2_1"/>
<font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:37_5_1_1_2"/><i><font color="Green" title="E88">A342</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="#E1:37_5_1_1_2"><i><font color="Green" title="E83">A330</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2"><i><font color="Green" title="E86">A333</font></i></a></i>;<br/><a NAME="E8:37_5_1_1_2"/><i><font color="Green" title="E89">A343</font></i>: 
<font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="#E2:37_5_1"><i><font color="Green" title="E82">A321</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2"><i><font color="Green" title="E83">A330</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2"><i><font color="Green" title="E86">A333</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c26">ZZ</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="#E6:37_5_1_1_2"><i><font color="Green" title="E87">A339</font></i></a>, <a class="txt" href="#E7:37_5_1_1_2"><i><font color="Green" title="E88">A342</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c27">Z1</font> = <font color="Maroon" title="c24">Z</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c26">ZZ</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c27">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )</span><br/><a NAME="E10:37_5_1_1_2"/><i><font color="Green" title="E90">A344</font></i>: 
<font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E2:37_5_1"><i><font color="Green" title="E82">A321</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2"><i><font color="Green" title="E83">A330</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2"><i><font color="Green" title="E86">A333</font></i></a></i>;<br/><b>hence </b><a NAME="E11:37_5_1_1_2"/>
<font color="Maroon" title="c27">Z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="#E3:37_5_1_1_2"><i><font color="Green" title="E84">A331</font></i></a>, <a class="txt" href="#E7:37_5_1_1_2"><i><font color="Green" title="E88">A342</font></i></a>, <a class="ref" href="xxreal_1.html#T165" title="XXREAL_1:th.165">XXREAL_1:165</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span><br/><a NAME="E12:37_5_1_1_2"/><i><font color="Green" title="E91">A345</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E10:37_5"><i><font color="Green" title="E61">A300</font></i></a>, <a class="txt" href="#E11:37_5"><i><font color="Green" title="E62">A301</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E13:37_5_1_1_2"/><i><font color="Green" title="E92">A346</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E12:37_5"><i><font color="Green" title="E63">A302</font></i></a>, <a class="txt" href="#E25:37_5"><i><font color="Green" title="E73">A312</font></i></a>, <a class="txt" href="#E26:37_5"><i><font color="Green" title="E74">A313</font></i></a>, <a class="txt" href="#E27:37_5"><i><font color="Green" title="E75">A314</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/><a NAME="E14:37_5_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c28">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font> )</span><br/>

<b>assume </b><a NAME="E1:37_5_1_1_2_3"/><i><font color="Green" title="E93">A347</font></i>: 
<font color="Maroon" title="c28">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font></span><br/>

<b>consider </b><font color="Maroon" title="c29">x1</font>, <font color="Maroon" title="c30">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_5_1_1_2_3"/><i><font color="Green" title="E94">A348</font></i>: 
<font color="Maroon" title="c29">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E4:37_5_1_1_2_3"/><i><font color="Green" title="E95">A349</font></i>: 
<font color="Maroon" title="c30">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">Z1</font>
 <b>and </b><br/><a NAME="E5:37_5_1_1_2_3"/><i><font color="Green" title="E96">A350</font></i>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<font color="Maroon" title="c30">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E12:37_5_1_1_2"><i><font color="Green" title="E91">A345</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3"><i><font color="Green" title="E93">A347</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c31">xy</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c29">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font> <b>by </b><i><a class="txt" href="#E3:37_5_1_1_2_3"><i><font color="Green" title="E94">A348</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:37_5_1_1_2_3"/><i><font color="Green" title="E97">A351</font></i>: 
<font color="Maroon" title="c31">xy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">N2</font>
 
<b>by </b><i><a class="txt" href="#E12:37_5"><i><font color="Green" title="E63">A302</font></i></a>, <a class="txt" href="#E3:37_5_1_1_2_3"><i><font color="Green" title="E94">A348</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:37_5_1_1_2_3"/><i><font color="Green" title="E98">A352</font></i>: 
<font color="Maroon" title="c31">xy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">N1</font>
 
<b>by </b><i><a class="txt" href="#E10:37_5"><i><font color="Green" title="E61">A300</font></i></a>, <a class="txt" href="#E3:37_5_1_1_2_3"><i><font color="Green" title="E94">A348</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E9:37_5_1_1_2_3"/><i><font color="Green" title="E99">A353</font></i>: 
<font color="Maroon" title="c29">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font>
 
<b>by </b><i><a class="txt" href="#E3:37_5_1_1_2_3"><i><font color="Green" title="E94">A348</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E100">A354</font></i>: 
<font color="Maroon" title="c31">xy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">N</font>
 
<b>by </b><i><a class="txt" href="#E20:37_5"><i><font color="Green" title="E69">A308</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c32">xZZ</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N1</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="txt" href="#E28:37_5"><i><font color="Green" title="E76">A315</font></i></a>, <a class="txt" href="#E29:37_5"><i><font color="Green" title="E77">A316</font></i></a>, <a class="txt" href="#E8:37_5_1_1_2_3"><i><font color="Green" title="E98">A352</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c33">Ui</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E13:37_5_1_1_2_3"/><i><font color="Green" title="E101">A355</font></i>: 
<font color="Maroon" title="c33">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">UL</font>
 <b>and </b><br/><a NAME="E14:37_5_1_1_2_3"/><i><font color="Green" title="E102">A356</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c33">Ui</font>
 <b>by </b><i><a class="txt" href="#E21:37_5"><i><font color="Green" title="E70">A309</font></i></a>, <a class="txt" href="#E2:37_5_1"><i><font color="Green" title="E82">A321</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2"><i><font color="Green" title="E83">A330</font></i></a></i>;<br/>
<a NAME="E15:37_5_1_1_2_3"/><i><font color="Green" title="E103">A357</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c33">Ui</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">Ui</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c34">D</font> being  <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a> <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <b> such that </b><br/><a NAME="E17:37_5_1_1_2_3"/><i><font color="Green" title="E104">A358</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c34">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c33">Ui</font>
 <b>and </b><br/><a NAME="E18:37_5_1_1_2_3"/><i><font color="Green" title="E105">A359</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">D</font> holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c33">Ui</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Olive" title="b2">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 <b>by </b><i><a class="txt" href="#E10:37"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="#E13:37_5_1_1_2_3"><i><font color="Green" title="E101">A355</font></i></a></i>;<br/>
<a NAME="E19:37_5_1_1_2_3"/><i><font color="Green" title="E106">A360</font></i>: 
<font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Z</font>
 
<b>by </b><i><a class="txt" href="#E3:37_5_1_1_2"><i><font color="Green" title="E84">A331</font></i></a>, <a class="txt" href="#E7:37_5_1_1_2"><i><font color="Green" title="E88">A342</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E20:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E107">A361</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:37_5_1_1_2_3"><i><font color="Green" title="E94">A348</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E21:37_5_1_1_2_3"/><i><font color="Green" title="E108">A362</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E12:37_5"><i><font color="Green" title="E63">A302</font></i></a>, <a class="txt" href="#E3:37_5_1_1_2_3"><i><font color="Green" title="E94">A348</font></i></a>, <a class="txt" href="#E19:37_5_1_1_2_3"><i><font color="Green" title="E106">A360</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:37_5_1_1_2_3"/><i><font color="Green" title="E109">A363</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E23:37_5_1_1_2_3"/><i><font color="Green" title="E110">A364</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N1</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E10:37_5"><i><font color="Green" title="E61">A300</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>then </b><i><font color="Green" title="E111">A365</font></i>: <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E8:37_5"><i><font color="Green" title="E59">A298</font></i></a>, <a class="txt" href="#E20:37_5_1_1_2_3"><i><font color="Green" title="E107">A361</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E11:37_5"><i><font color="Green" title="E62">A301</font></i></a>, <a class="txt" href="#E20:37_5_1_1_2_3"><i><font color="Green" title="E107">A361</font></i></a>, <a class="txt" href="#E23:37_5_1_1_2_3"><i><font color="Green" title="E110">A364</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E25:37_5_1_1_2_3"/><i><font color="Green" title="E112">A366</font></i>: 
<font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">ZZ</font>
 
<b>by </b><i><a class="txt" href="#E10:37_5_1_1_2"><i><font color="Green" title="E90">A344</font></i></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E26:37_5_1_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E20:37_5"><i><font color="Green" title="E69">A308</font></i></a>, <a class="txt" href="#E9:37_5_1_1_2_3"><i><font color="Green" title="E99">A353</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E27:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E113">A367</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E28:37_5_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c33">Ui</font>
 
<b>by </b><i><a class="txt" href="#E14:37_5_1_1_2_3"><i><font color="Green" title="E102">A356</font></i></a>, <a class="txt" href="#E24:37_5_1_1_2_3"><i><font color="Green" title="E111">A365</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c35">Uit</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E30:37_5_1_1_2_3"/><i><font color="Green" title="E114">A368</font></i>: 
<font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">Uit</font>
 <b>and </b><br/><a NAME="E31:37_5_1_1_2_3"/><i><font color="Green" title="E115">A369</font></i>: 
<font color="Maroon" title="c35">Uit</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">D</font>
 <b>by </b><i><a class="txt" href="#E17:37_5_1_1_2_3"><i><font color="Green" title="E104">A358</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c36">Uit</font> = <font color="Maroon" title="c35">Uit</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="#E30:37_5_1_1_2_3"><i><font color="Green" title="E114">A368</font></i></a>, <a class="txt" href="#E31:37_5_1_1_2_3"><i><font color="Green" title="E115">A369</font></i></a></i>;<br/>
<a NAME="E33:37_5_1_1_2_3"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font> &amp; <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E2:37_5_1"><i><font color="Green" title="E82">A321</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2"><i><font color="Green" title="E83">A330</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2"><i><font color="Green" title="E86">A333</font></i></a></i>;<br/>
<a NAME="E34:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E116">A370</font></i>: 
<font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<a NAME="E35:37_5_1_1_2_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:37_5_1"><i><font color="Green" title="E82">A321</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2"><i><font color="Green" title="E86">A333</font></i></a></i>;<br/>
<a NAME="E36:37_5_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="#E8:37_5_1_1_2"><i><font color="Green" title="E89">A343</font></i></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<a NAME="E37:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E117">A371</font></i>: 
<font color="Maroon" title="c26">ZZ</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="#E10:37_5_1_1_2"><i><font color="Green" title="E90">A344</font></i></a>, <a class="txt" href="#E34:37_5_1_1_2_3"><i><font color="Green" title="E116">A370</font></i></a>, <a class="ref" href="borsuk_4.html#T49" title="BORSUK_4:th.49">BORSUK_4:49</a></i>;<br/>
<a NAME="E38:37_5_1_1_2_3"/>
<font color="Maroon" title="c31">xy</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="#E3:37_5_1_1_2_3"><i><font color="Green" title="E94">A348</font></i></a>, <a class="ref" href="connsp_1.html#T29" title="CONNSP_1:th.29">CONNSP_1:29</a></i>;<br/>
<a NAME="E39:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E118">A372</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="#E37:37_5_1_1_2_3"><i><font color="Green" title="E117">A371</font></i></a>, <a class="ref" href="topalg_3.html#T17" title="TOPALG_3:th.17">TOPALG_3:17</a></i>;<br/>
<a NAME="E40:37_5_1_1_2_3"/><i><font color="Green" title="E119">A373</font></i>: 
 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2" title="TSEP_1:th.2">TSEP_1:2</a></i>;<br/>
<a NAME="E41:37_5_1_1_2_3"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N1</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T25" title="BORSUK_3:th.25">BORSUK_3:25</a></i>;<br/>
<a NAME="E42:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E120">A374</font></i>: 
<font color="Maroon" title="c32">xZZ</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="#E39:37_5_1_1_2_3"><i><font color="Green" title="E118">A372</font></i></a>, <a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/>
<a NAME="E43:37_5_1_1_2_3"/><i><font color="Green" title="E121">A375</font></i>: 
<font color="Maroon" title="c29">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c31">xy</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E44:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E122">A376</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E25:37_5_1_1_2_3"><i><font color="Green" title="E112">A366</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E45:37_5_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E46:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E123">A377</font></i>: 
<font color="Maroon" title="c36">Uit</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c18">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E30:37_5_1_1_2_3"><i><font color="Green" title="E114">A368</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E47:37_5_1_1_2_3"/>
<font color="Maroon" title="c34">D</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c18">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c37">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c37">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font> or <font color="Maroon" title="c37">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c34">D</font> )</span><br/>

<b>assume </b><a NAME="E1:37_5_1_1_2_3_2"/>
<font color="Maroon" title="c37">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c37">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c34">D</font></span><br/>

<b>then consider </b><font color="Maroon" title="c38">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N1</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E3:37_5_1_1_2_3_2"/><i><font color="Green" title="E124">A378</font></i>: 
<font color="Maroon" title="c38">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">xZZ</font>
 <b>and </b><br/><a NAME="E4:37_5_1_1_2_3_2"/><i><font color="Green" title="E125">A379</font></i>: 
<font color="Maroon" title="c18">Fn1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green" title="E126">A380</font></i>: <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Fn1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c38">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">a</font>
<b>by </b><i><a class="txt" href="#E8:37_5"><i><font color="Green" title="E59">A298</font></i></a>, <a class="txt" href="#E28:37_5"><i><font color="Green" title="E76">A315</font></i></a>, <a class="txt" href="#E29:37_5"><i><font color="Green" title="E77">A316</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:37_5_1_1_2_3_2"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E10:37_5_1_1_2_3"><i><font color="Green" title="E100">A354</font></i></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E7:37_5_1_1_2_3_2"/><b>then </b><i><font color="Green" title="E127">A381</font></i>: 
<font color="Maroon" title="c38">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:37_5_1_1_2_3_2"><i><font color="Green" title="E124">A378</font></i></a></i>;<br/>
<a NAME="E8:37_5_1_1_2_3_2"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E9:37_5_1_1_2_3_2"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E7:37_5_1_1_2_3_2"><i><font color="Green" title="E127">A381</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:37_5_1_1_2_3_2"/>
<font color="Maroon" title="c37">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c34">D</font>
 <b>by </b><i><a class="txt" href="#E14:37_5_1_1_2_3"><i><font color="Green" title="E102">A356</font></i></a>, <a class="txt" href="#E17:37_5_1_1_2_3"><i><font color="Green" title="E104">A358</font></i></a>, <a class="txt" href="#E4:37_5_1_1_2_3_2"><i><font color="Green" title="E125">A379</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2_3_2"><i><font color="Green" title="E126">A380</font></i></a>, <a class="txt" href="topalg_5.html#E35"><i><font color="Green" title="E29">Lm12</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E48:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E124">A382</font></i>: 
<font color="Maroon" title="c18">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c36">Uit</font>
 
<b>by </b><i><a class="txt" href="#E5:37_5"><i><font color="Green" title="E56">A295</font></i></a>, <a class="txt" href="#E31:37_5_1_1_2_3"><i><font color="Green" title="E115">A369</font></i></a>, <a class="txt" href="#E42:37_5_1_1_2_3"><i><font color="Green" title="E120">A374</font></i></a>, <a class="txt" href="#E46:37_5_1_1_2_3"><i><font color="Green" title="E123">A377</font></i></a>, <a class="ref" href="topalg_3.html#T13" title="TOPALG_3:th.13">TOPALG_3:13</a>, <a class="ref" href="tops_2.html#T75" title="TOPS_2:th.75">TOPS_2:75</a></i>;<br/>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">Fn2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E6:37_5"><i><font color="Green" title="E57">A296</font></i></a>, <a class="txt" href="#E21:37_5_1_1_2_3"><i><font color="Green" title="E108">A362</font></i></a>, <a class="txt" href="#E22:37_5_1_1_2_3"><i><font color="Green" title="E109">A363</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E25:37_5"><i><font color="Green" title="E73">A312</font></i></a>, <a class="txt" href="#E26:37_5"><i><font color="Green" title="E74">A313</font></i></a>, <a class="txt" href="#E27:37_5"><i><font color="Green" title="E75">A314</font></i></a>, <a class="txt" href="#E21:37_5_1_1_2_3"><i><font color="Green" title="E108">A362</font></i></a>, <a class="txt" href="#E22:37_5_1_1_2_3"><i><font color="Green" title="E109">A363</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E50:37_5_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c33">Ui</font>
 
<b>by </b><i><a class="txt" href="#E14:37_5_1_1_2_3"><i><font color="Green" title="E102">A356</font></i></a>, <a class="txt" href="#E27:37_5_1_1_2_3"><i><font color="Green" title="E113">A367</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c37">Uith</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E52:37_5_1_1_2_3"/><i><font color="Green" title="E125">A383</font></i>: 
<font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c37">Uith</font>
 <b>and </b><br/><a NAME="E53:37_5_1_1_2_3"/><i><font color="Green" title="E126">A384</font></i>: 
<font color="Maroon" title="c37">Uith</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">D</font>
 <b>by </b><i><a class="txt" href="#E17:37_5_1_1_2_3"><i><font color="Green" title="E104">A358</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c38">Uith</font> = <font color="Maroon" title="c37">Uith</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="#E52:37_5_1_1_2_3"><i><font color="Green" title="E125">A383</font></i></a>, <a class="txt" href="#E53:37_5_1_1_2_3"><i><font color="Green" title="E126">A384</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c39">XZZ</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="txt" href="#E26:37_5"><i><font color="Green" title="E74">A313</font></i></a>, <a class="txt" href="#E27:37_5"><i><font color="Green" title="E75">A314</font></i></a>, <a class="txt" href="#E7:37_5_1_1_2_3"><i><font color="Green" title="E97">A351</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E56:37_5_1_1_2_3"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E40:37_5_1_1_2_3"><i><font color="Green" title="E119">A373</font></i></a>, <a class="ref" href="borsuk_3.html#T25" title="BORSUK_3:th.25">BORSUK_3:25</a></i>;<br/>
<a NAME="E57:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E127">A385</font></i>: 
<font color="Maroon" title="c39">XZZ</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="#E39:37_5_1_1_2_3"><i><font color="Green" title="E118">A372</font></i></a>, <a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/>
<a NAME="E58:37_5_1_1_2_3"/><i><font color="Green" title="E128">A386</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N2</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E7:37_5_1_1_2_3"><i><font color="Green" title="E97">A351</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E59:37_5_1_1_2_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:37_5_1_1_2_3"><i><font color="Green" title="E94">A348</font></i></a>, <a class="txt" href="#E9:37_5_1_1_2_3"><i><font color="Green" title="E99">A353</font></i></a>, <a class="txt" href="#E25:37_5_1_1_2_3"><i><font color="Green" title="E112">A366</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E60:37_5_1_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c19">Fn2</font>
 
<b>by </b><i><a class="txt" href="#E13:37_5"><i><font color="Green" title="E64">A303</font></i></a>, <a class="txt" href="#E58:37_5_1_1_2_3"><i><font color="Green" title="E128">A386</font></i></a></i>;<br/>
<a NAME="E61:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E129">A387</font></i>: 
<font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Fn2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E44:37_5_1_1_2_3"><i><font color="Green" title="E122">A376</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E62:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E130">A388</font></i>: 
<font color="Maroon" title="c38">Uith</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c19">Fn2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E52:37_5_1_1_2_3"><i><font color="Green" title="E125">A383</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E63:37_5_1_1_2_3"/>
<font color="Maroon" title="c34">D</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c19">Fn2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_2_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c40">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c40">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Fn2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font> or <font color="Maroon" title="c40">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c34">D</font> )</span><br/>

<b>assume </b><a NAME="E1:37_5_1_1_2_3_4"/>
<font color="Maroon" title="c40">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Fn2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c40">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c34">D</font></span><br/>

<b>then consider </b><font color="Maroon" title="c41">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N2</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E3:37_5_1_1_2_3_4"/><i><font color="Green" title="E131">A389</font></i>: 
<font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">xZZ</font>
 <b>and </b><br/><a NAME="E4:37_5_1_1_2_3_4"/><i><font color="Green" title="E132">A390</font></i>: 
<font color="Maroon" title="c19">Fn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c41">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green" title="E133">A391</font></i>: <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c41">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">Fn2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c41">a</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">a</font>
<b>by </b><i><a class="txt" href="#E6:37_5"><i><font color="Green" title="E57">A296</font></i></a>, <a class="txt" href="#E26:37_5"><i><font color="Green" title="E74">A313</font></i></a>, <a class="txt" href="#E27:37_5"><i><font color="Green" title="E75">A314</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:37_5_1_1_2_3_4"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E10:37_5_1_1_2_3"><i><font color="Green" title="E100">A354</font></i></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E7:37_5_1_1_2_3_4"/><b>then </b><i><font color="Green" title="E134">A392</font></i>: 
<font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:37_5_1_1_2_3_4"><i><font color="Green" title="E131">A389</font></i></a></i>;<br/>
<a NAME="E8:37_5_1_1_2_3_4"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E9:37_5_1_1_2_3_4"/><b>then </b>
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c41">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c22">N</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E11:37"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E7:37_5_1_1_2_3_4"><i><font color="Green" title="E134">A392</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:37_5_1_1_2_3_4"/>
<font color="Maroon" title="c40">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c34">D</font>
 <b>by </b><i><a class="txt" href="#E14:37_5_1_1_2_3"><i><font color="Green" title="E102">A356</font></i></a>, <a class="txt" href="#E17:37_5_1_1_2_3"><i><font color="Green" title="E104">A358</font></i></a>, <a class="txt" href="#E4:37_5_1_1_2_3_4"><i><font color="Green" title="E132">A390</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2_3_4"><i><font color="Green" title="E133">A391</font></i></a>, <a class="txt" href="topalg_5.html#E35"><i><font color="Green" title="E29">Lm12</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E64:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E131">A393</font></i>: 
<font color="Maroon" title="c19">Fn2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c38">Uith</font>
 
<b>by </b><i><a class="txt" href="#E4:37_5"><i><font color="Green" title="E55">A294</font></i></a>, <a class="txt" href="#E53:37_5_1_1_2_3"><i><font color="Green" title="E126">A384</font></i></a>, <a class="txt" href="#E57:37_5_1_1_2_3"><i><font color="Green" title="E127">A385</font></i></a>, <a class="txt" href="#E62:37_5_1_1_2_3"><i><font color="Green" title="E130">A388</font></i></a>, <a class="ref" href="topalg_3.html#T13" title="TOPALG_3:th.13">TOPALG_3:13</a>, <a class="ref" href="tops_2.html#T75" title="TOPS_2:th.75">TOPS_2:75</a></i>;<br/>
<font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E20:37_5_1_1_2_3"><i><font color="Green" title="E107">A361</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E4:37_5_1_1_2"><i><font color="Green" title="E85">A332</font></i></a>, <a class="txt" href="#E20:37_5_1_1_2_3"><i><font color="Green" title="E107">A361</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E66:37_5_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c36">Uit</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c38">Uith</font>
 
<b>by </b><i><a class="txt" href="#E30:37_5_1_1_2_3"><i><font color="Green" title="E114">A368</font></i></a>, <a class="txt" href="#E61:37_5_1_1_2_3"><i><font color="Green" title="E129">A387</font></i></a>, <a class="txt" href="#E64:37_5_1_1_2_3"><i><font color="Green" title="E131">A393</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E67:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E132">A394</font></i>: 
<font color="Maroon" title="c36">Uit</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">Uith</font>
 
<b>by </b><i><a class="txt" href="#E31:37_5_1_1_2_3"><i><font color="Green" title="E115">A369</font></i></a>, <a class="txt" href="#E53:37_5_1_1_2_3"><i><font color="Green" title="E126">A384</font></i></a>, <a class="ref" href="taxonom2.html#D5" title="TAXONOM2:def.5">TAXONOM2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c40">f</font> = <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c36">Uit</font>;<br/>
<a NAME="E68:37_5_1_1_2_3"/><i><font color="Green" title="E133">A395</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c36">Uit</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">Uit</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E69:37_5_1_1_2_3"/><i><font color="Green" title="E134">A396</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c36">Uit</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">Uit</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green" title="E29">Lm12</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E70:37_5_1_1_2_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c36">Uit</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c33">Ui</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_2_3_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c41">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c41">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c36">Uit</font></span>)</span> or <font color="Maroon" title="c41">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Ui</font> )</span><br/>

<b>assume </b><a NAME="E1:37_5_1_1_2_3_6"/>
<font color="Maroon" title="c41">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c36">Uit</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c41">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Ui</font></span><br/>

<b>then consider </b><font color="Maroon" title="c42">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_5_1_1_2_3_6"/><i><font color="Green" title="E135">A397</font></i>: 
<font color="Maroon" title="c42">a</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> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c36">Uit</font></span>)</span>
 <b>and </b><br/><a NAME="E4:37_5_1_1_2_3_6"/><i><font color="Green" title="E136">A398</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c36">Uit</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:37_5_1_1_2_3_6"/>
<font color="Maroon" title="c42">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c34">D</font>
 
<b>by </b><i><a class="txt" href="#E31:37_5_1_1_2_3"><i><font color="Green" title="E115">A369</font></i></a>, <a class="txt" href="#E69:37_5_1_1_2_3"><i><font color="Green" title="E134">A396</font></i></a>, <a class="txt" href="#E3:37_5_1_1_2_3_6"><i><font color="Green" title="E135">A397</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:37_5_1_1_2_3_6"/><b>then </b>
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Ui</font>
 
<b>by </b><i><a class="txt" href="#E17:37_5_1_1_2_3"><i><font color="Green" title="E104">A358</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E7:37_5_1_1_2_3_6"/>
<font color="Maroon" title="c41">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Ui</font>
 <b>by </b><i><a class="txt" href="#E69:37_5_1_1_2_3"><i><font color="Green" title="E134">A396</font></i></a>, <a class="txt" href="#E3:37_5_1_1_2_3_6"><i><font color="Green" title="E135">A397</font></i></a>, <a class="txt" href="#E4:37_5_1_1_2_3_6"><i><font color="Green" title="E136">A398</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c41">f</font> = <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c36">Uit</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c36">Uit</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c33">Ui</font></span>)</span> <b>by </b><i><a class="txt" href="#E15:37_5_1_1_2_3"><i><font color="Green" title="E103">A357</font></i></a>, <a class="txt" href="#E68:37_5_1_1_2_3"><i><font color="Green" title="E133">A395</font></i></a>, <a class="txt" href="#E69:37_5_1_1_2_3"><i><font color="Green" title="E134">A396</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E72:37_5_1_1_2_3"/>
<font color="Maroon" title="c41">f</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>
 
<b>by </b><i><a class="txt" href="#E18:37_5_1_1_2_3"><i><font color="Green" title="E105">A359</font></i></a>, <a class="txt" href="#E31:37_5_1_1_2_3"><i><font color="Green" title="E115">A369</font></i></a></i>;<br/>
<a NAME="E73:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E135">A399</font></i>: 
<font color="Maroon" title="c41">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E74:37_5_1_1_2_3"/><i><font color="Green" title="E136">A400</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E28:37_5"><i><font color="Green" title="E76">A315</font></i></a>, <a class="txt" href="#E29:37_5"><i><font color="Green" title="E77">A316</font></i></a>, <a class="txt" href="#E30:37_5"><i><font color="Green" title="E78">A317</font></i></a>, <a class="txt" href="#E8:37_5_1_1_2_3"><i><font color="Green" title="E98">A352</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E75:37_5_1_1_2_3"/><i><font color="Green" title="E137">A401</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E25:37_5"><i><font color="Green" title="E73">A312</font></i></a>, <a class="txt" href="#E26:37_5"><i><font color="Green" title="E74">A313</font></i></a>, <a class="txt" href="#E27:37_5"><i><font color="Green" title="E75">A314</font></i></a>, <a class="txt" href="#E7:37_5_1_1_2_3"><i><font color="Green" title="E97">A351</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E76:37_5_1_1_2_3"/><i><font color="Green" title="E138">A402</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c41">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_2_3_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c42">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> or <font color="Maroon" title="c42">b</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="c41">f</font> )</span><br/>

<b>assume </b><a NAME="E1:37_5_1_1_2_3_7"/>
<font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c42">b</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="c41">f</font></span><br/>

<b>then consider </b><font color="Maroon" title="c43">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_5_1_1_2_3_7"/><i><font color="Green" title="E139">A403</font></i>: 
<font color="Maroon" title="c43">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:37_5_1_1_2_3_7"/><i><font color="Green" title="E140">A404</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c43">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:37_5_1_1_2_3_7"/>
<font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c43">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">b</font>
 
<b>by </b><i><a class="txt" href="#E74:37_5_1_1_2_3"><i><font color="Green" title="E136">A400</font></i></a>, <a class="txt" href="#E3:37_5_1_1_2_3_7"><i><font color="Green" title="E139">A403</font></i></a>, <a class="txt" href="#E4:37_5_1_1_2_3_7"><i><font color="Green" title="E140">A404</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:37_5_1_1_2_3_7"/><b>then </b>
<font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E30:37_5"><i><font color="Green" title="E78">A317</font></i></a>, <a class="txt" href="#E74:37_5_1_1_2_3"><i><font color="Green" title="E136">A400</font></i></a>, <a class="txt" href="#E3:37_5_1_1_2_3_7"><i><font color="Green" title="E139">A403</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:37_5_1_1_2_3_7"/>
<font color="Maroon" title="c42">b</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="c41">f</font>
 <b>by </b><i><a class="txt" href="#E48:37_5_1_1_2_3"><i><font color="Green" title="E124">A382</font></i></a>, <a class="txt" href="#E69:37_5_1_1_2_3"><i><font color="Green" title="E134">A396</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E77:37_5_1_1_2_3"/><i><font color="Green" title="E139">A405</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c41">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_2_3_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c42">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> or <font color="Maroon" title="c42">b</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="c41">f</font> )</span><br/>

<b>assume </b><a NAME="E1:37_5_1_1_2_3_8"/>
<font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c42">b</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="c41">f</font></span><br/>

<b>then consider </b><font color="Maroon" title="c43">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:37_5_1_1_2_3_8"/><i><font color="Green" title="E140">A406</font></i>: 
<font color="Maroon" title="c43">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:37_5_1_1_2_3_8"/><i><font color="Green" title="E141">A407</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c43">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">b</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:37_5_1_1_2_3_8"/>
<font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c43">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">b</font>
 
<b>by </b><i><a class="txt" href="#E75:37_5_1_1_2_3"><i><font color="Green" title="E137">A401</font></i></a>, <a class="txt" href="#E3:37_5_1_1_2_3_8"><i><font color="Green" title="E140">A406</font></i></a>, <a class="txt" href="#E4:37_5_1_1_2_3_8"><i><font color="Green" title="E141">A407</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:37_5_1_1_2_3_8"/><b>then </b>
<font color="Maroon" title="c42">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Fn2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c32">xZZ</font>
 
<b>by </b><i><a class="txt" href="#E25:37_5"><i><font color="Green" title="E73">A312</font></i></a>, <a class="txt" href="#E75:37_5_1_1_2_3"><i><font color="Green" title="E137">A401</font></i></a>, <a class="txt" href="#E3:37_5_1_1_2_3_8"><i><font color="Green" title="E140">A406</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:37_5_1_1_2_3_8"/>
<font color="Maroon" title="c42">b</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="c41">f</font>
 <b>by </b><i><a class="txt" href="#E64:37_5_1_1_2_3"><i><font color="Green" title="E131">A393</font></i></a>, <a class="txt" href="#E67:37_5_1_1_2_3"><i><font color="Green" title="E132">A394</font></i></a>, <a class="txt" href="#E69:37_5_1_1_2_3"><i><font color="Green" title="E134">A396</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E78:37_5_1_1_2_3"/><i><font color="Green" title="E140">A408</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E76:37_5_1_1_2_3"><i><font color="Green" title="E138">A402</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E79:37_5_1_1_2_3"/><i><font color="Green" title="E141">A409</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E77:37_5_1_1_2_3"><i><font color="Green" title="E139">A405</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E80:37_5_1_1_2_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_5_1_1_2_3_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c42">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c42">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font> )</span><br/>

<b>assume </b><a NAME="E1:37_5_1_1_2_3_9"/><i><font color="Green" title="E142">A410</font></i>: 
<font color="Maroon" title="c42">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font></span><br/>

<a NAME="E2:37_5_1_1_2_3_9"/><i><font color="Green" title="E143">A411</font></i>: 
<font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Fn1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E30:37_5"><i><font color="Green" title="E78">A317</font></i></a>, <a class="txt" href="#E74:37_5_1_1_2_3"><i><font color="Green" title="E136">A400</font></i></a>, <a class="txt" href="#E78:37_5_1_1_2_3"><i><font color="Green" title="E140">A408</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3_9"><i><font color="Green" title="E142">A410</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E3:37_5_1_1_2_3_9"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">Fn1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E4:37_5_1_1_2_3_9"/><b>then </b><i><font color="Green" title="E144">A412</font></i>: 
<font color="Maroon" title="c42">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E78:37_5_1_1_2_3"><i><font color="Green" title="E140">A408</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3_9"><i><font color="Green" title="E142">A410</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:37_5_1_1_2_3_9"/><i><font color="Green" title="E145">A413</font></i>: 
<font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Fn2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E13:37_5"><i><font color="Green" title="E64">A303</font></i></a>, <a class="txt" href="#E58:37_5_1_1_2_3"><i><font color="Green" title="E128">A386</font></i></a>, <a class="txt" href="#E74:37_5_1_1_2_3"><i><font color="Green" title="E136">A400</font></i></a>, <a class="txt" href="#E78:37_5_1_1_2_3"><i><font color="Green" title="E140">A408</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3_9"><i><font color="Green" title="E142">A410</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:37_5_1_1_2_3_9"/><i><font color="Green" title="E146">A414</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">Fn1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Fn1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="ref" href="relat_1.html#T112" title="RELAT_1:th.112">RELAT_1:112</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="txt" href="#E74:37_5_1_1_2_3"><i><font color="Green" title="E136">A400</font></i></a>, <a class="txt" href="#E78:37_5_1_1_2_3"><i><font color="Green" title="E140">A408</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3_9"><i><font color="Green" title="E142">A410</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c41">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E30:37_5"><i><font color="Green" title="E78">A317</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3_9"><i><font color="Green" title="E142">A410</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E48:37_5_1_1_2_3"><i><font color="Green" title="E124">A382</font></i></a>, <a class="txt" href="#E2:37_5_1_1_2_3_9"><i><font color="Green" title="E143">A411</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Fn1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="txt" href="#E30:37_5"><i><font color="Green" title="E78">A317</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3_9"><i><font color="Green" title="E142">A410</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="txt" href="#E8:37_5"><i><font color="Green" title="E59">A298</font></i></a>, <a class="txt" href="#E11:37_5"><i><font color="Green" title="E62">A301</font></i></a>, <a class="txt" href="#E78:37_5_1_1_2_3"><i><font color="Green" title="E140">A408</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3_9"><i><font color="Green" title="E142">A410</font></i></a>, <a class="txt" href="#E6:37_5_1_1_2_3_9"><i><font color="Green" title="E146">A414</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">Fn2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="txt" href="#E6:37_5"><i><font color="Green" title="E57">A296</font></i></a>, <a class="txt" href="#E58:37_5_1_1_2_3"><i><font color="Green" title="E128">A386</font></i></a>, <a class="txt" href="#E4:37_5_1_1_2_3_9"><i><font color="Green" title="E144">A412</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E13:37_5"><i><font color="Green" title="E64">A303</font></i></a>, <a class="txt" href="#E58:37_5_1_1_2_3"><i><font color="Green" title="E128">A386</font></i></a>, <a class="txt" href="#E4:37_5_1_1_2_3_9"><i><font color="Green" title="E144">A412</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c41">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c42">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E64:37_5_1_1_2_3"><i><font color="Green" title="E131">A393</font></i></a>, <a class="txt" href="#E67:37_5_1_1_2_3"><i><font color="Green" title="E132">A394</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2_3_9"><i><font color="Green" title="E145">A413</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">Fn2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="txt" href="#E13:37_5"><i><font color="Green" title="E64">A303</font></i></a>, <a class="txt" href="#E58:37_5_1_1_2_3"><i><font color="Green" title="E128">A386</font></i></a>, <a class="txt" href="#E4:37_5_1_1_2_3_9"><i><font color="Green" title="E144">A412</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">Fn2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="txt" href="#E74:37_5_1_1_2_3"><i><font color="Green" title="E136">A400</font></i></a>, <a class="txt" href="#E78:37_5_1_1_2_3"><i><font color="Green" title="E140">A408</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3_9"><i><font color="Green" title="E142">A410</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c42">x</font>
<b>by </b><i><a class="ref" href="relat_1.html#T112" title="RELAT_1:th.112">RELAT_1:112</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E81:37_5_1_1_2_3"/><b>then </b><i><font color="Green" title="E142">A415</font></i>: 
<font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E74:37_5_1_1_2_3"><i><font color="Green" title="E136">A400</font></i></a>, <a class="txt" href="#E75:37_5_1_1_2_3"><i><font color="Green" title="E137">A401</font></i></a>, <a class="txt" href="#E78:37_5_1_1_2_3"><i><font color="Green" title="E140">A408</font></i></a>, <a class="txt" href="#E79:37_5_1_1_2_3"><i><font color="Green" title="E141">A409</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_2_3_10"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_5_1_1_2_3_10"/>
( <font color="Maroon" title="c30">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">ZZ</font> or  not <font color="Maroon" title="c30">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">ZZ</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_2_3_10_1"><b>suppose </b></a><a NAME="E1:37_5_1_1_2_3_10_1"/>
<font color="Maroon" title="c30">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">ZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font></span><br/><div class="add"><a NAME="E2:37_5_1_1_2_3_10_1"/><b>then </b><i><font color="Green" title="E143">A416</font></i>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E5:37_5_1_1_2_3"><i><font color="Green" title="E96">A350</font></i></a>, <a class="txt" href="#E43:37_5_1_1_2_3"><i><font color="Green" title="E121">A375</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font> = 
<font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
<b>by </b><i><a class="txt" href="#E12:37_5_1_1_2"><i><font color="Green" title="E91">A345</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3"><i><font color="Green" title="E93">A347</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
<b>by </b><i><a class="txt" href="#E2:37_5_1_1_2_3_10_1"><i><font color="Green" title="E143">A416</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c31">xy</font>,<font color="Maroon" title="c26">ZZ</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
<b>by </b><i><a class="txt" href="#E73:37_5_1_1_2_3"><i><font color="Green" title="E135">A399</font></i></a>, <a class="txt" href="#E74:37_5_1_1_2_3"><i><font color="Green" title="E136">A400</font></i></a>, <a class="txt" href="#E75:37_5_1_1_2_3"><i><font color="Green" title="E137">A401</font></i></a>, <a class="txt" href="#E76:37_5_1_1_2_3"><i><font color="Green" title="E138">A402</font></i></a>, <a class="txt" href="#E77:37_5_1_1_2_3"><i><font color="Green" title="E139">A405</font></i></a>, <a class="txt" href="#E81:37_5_1_1_2_3"><i><font color="Green" title="E142">A415</font></i></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></i>
<br/>.= 
<font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
<b>by </b><i><a class="txt" href="#E2:37_5_1_1_2_3_10_1"><i><font color="Green" title="E143">A416</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
<b>by </b><i><a class="txt" href="#E12:37_5_1_1_2"><i><font color="Green" title="E91">A345</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3"><i><font color="Green" title="E93">A347</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_5_1_1_2_3_10_2"><b>suppose </b></a><a NAME="E1:37_5_1_1_2_3_10_2"/>
not <font color="Maroon" title="c30">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">ZZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font></span><br/><div class="add"><a NAME="E2:37_5_1_1_2_3_10_2"/><b>then </b>
<font color="Maroon" title="c30">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Z</font>
 <b>by </b><i><a class="txt" href="#E4:37_5_1_1_2_3"><i><font color="Green" title="E95">A349</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:37_5_1_1_2_3_10_2"/><b>then </b><i><font color="Green" title="E143">A417</font></i>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E3:37_5_1_1_2_3"><i><font color="Green" title="E94">A348</font></i></a>, <a class="txt" href="#E5:37_5_1_1_2_3"><i><font color="Green" title="E96">A350</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font> = 
<font color="Maroon" title="c18">Fn1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
<b>by </b><i><a class="txt" href="#E12:37_5_1_1_2"><i><font color="Green" title="E91">A345</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3"><i><font color="Green" title="E93">A347</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c24">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
<b>by </b><i><a class="txt" href="#E3:37_5_1_1_2_3_10_2"><i><font color="Green" title="E143">A417</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c19">Fn2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
<b>by </b><i><a class="txt" href="#E4:37_5_1_1_2"><i><font color="Green" title="E85">A332</font></i></a>, <a class="txt" href="#E3:37_5_1_1_2_3_10_2"><i><font color="Green" title="E143">A417</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
<b>by </b><i><a class="txt" href="#E12:37_5_1_1_2"><i><font color="Green" title="E91">A345</font></i></a>, <a class="txt" href="#E1:37_5_1_1_2_3"><i><font color="Green" title="E93">A347</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E15:37_5_1_1_2"/>
<font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c27">Z1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="#E12:37_5_1_1_2"><i><font color="Green" title="E91">A345</font></i></a>, <a class="txt" href="#E13:37_5_1_1_2"><i><font color="Green" title="E92">A346</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E33:37_5"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E31:37_5"><i><font color="Green" title="E79">A318</font></i></a>, <a class="txt" href="#E32:37_5"><i><font color="Green" title="E80">A319</font></i></a>);<br/></i>
<a NAME="E34:37_5"/><b>then </b>
 ex <font color="Olive" title="b1">Z</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c20">TT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">TT</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Olive" title="b1">Z</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )
 
<b>by </b><i><a class="txt" href="#E24:37_5"><i><font color="Green" title="E72">A311</font></i></a></i>;<br/>
<b>hence </b><a NAME="E35:37_5"/>
<font color="Maroon" title="c18">Fn1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">Fn2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="#E16:37_5"><i><font color="Green" title="E66">A305</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E19:37"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">G</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> ex <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">G</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><b>let </b><font color="Maroon" title="c14">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font> &amp; <font color="Maroon" title="c14">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">G</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">V</font> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:37_6"/><i><font color="Green" title="E53">A418</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font>
 <b>and </b><br/><a NAME="E2:37_6"/><i><font color="Green" title="E54">A419</font></i>: 
<font color="Maroon" title="c14">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> st <br/>( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">G</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">V</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c15">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c16">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> , <font color="Maroon" title="c17">N</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c18">Fn</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <b> such that </b><br/><a NAME="E4:37_6"/><i><font color="Green" title="E55">A420</font></i>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:37_6"/><i><font color="Green" title="E56">A421</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">p</font>
 <b>and </b><br/><a NAME="E6:37_6"/><i><font color="Green" title="E57">A422</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font>
 <b>and </b><br/><a NAME="E7:37_6"/><i><font color="Green" title="E58">A423</font></i>: 
<font color="Maroon" title="c18">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E8:37_6"/><i><font color="Green" title="E59">A424</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Fn</font>
 <b>and </b><br/><a NAME="E9:37_6"/><i><font color="Green" title="E60">A425</font></i>: 
<font color="Maroon" title="c18">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E10:37_6"/>
for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c18">Fn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font>
 <b>by </b><i><a class="txt" href="#E14:37"><i><font color="Green" title="E48">A281</font></i></a></i>;<br/>
<i><font color="Green" title="E61">A426</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E12:37_6"/><b>then </b>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E4:37_6"><i><font color="Green" title="E55">A420</font></i></a>, <a class="txt" href="#E6:37_6"><i><font color="Green" title="E57">A422</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E14:37_6"/><i><font color="Green" title="E62">A427</font></i>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">W</font>
 <b>and </b><br/><a NAME="E15:37_6"/><i><font color="Green" title="E63">A428</font></i>: 
<font color="Maroon" title="c19">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E16:37_6"/><i><font color="Green" title="E64">A429</font></i>: 
<font color="Maroon" title="c18">Fn</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c19">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">V</font>
 <b>by </b><i><a class="txt" href="#E1:37_6"><i><font color="Green" title="E53">A418</font></i></a>, <a class="txt" href="#E2:37_6"><i><font color="Green" title="E54">A419</font></i></a>, <a class="txt" href="#E5:37_6"><i><font color="Green" title="E56">A421</font></i></a>, <a class="txt" href="#E7:37_6"><i><font color="Green" title="E58">A423</font></i></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E17:37_6"/><i><font color="Green" title="E65">A430</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">Fn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E11:37_6"><i><font color="Green" title="E61">A426</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:37_6"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E8"><i><font color="Green" title="E7">Lm7</font></i></a>, <a class="ref" href="borsuk_3.html#T26" title="BORSUK_3:th.26">BORSUK_3:26</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E20:37_6"/><i><font color="Green" title="E66">A431</font></i>: 
<font color="Maroon" title="c20">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E21:37_6"/><i><font color="Green" title="E67">A432</font></i>: 
<font color="Maroon" title="c20">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">W</font>
 <b>by </b><i><a class="txt" href="#E15:37_6"><i><font color="Green" title="E63">A428</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<b>take </b><font color="Maroon" title="c21">WW</font> = <font color="Maroon" title="c20">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">WW</font> &amp; <font color="Maroon" title="c21">WW</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">G</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c21">WW</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">V</font> )</span><br/>

<a NAME="E22:37_6"/><i><font color="Green" title="E68">A433</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">N</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/>
<a NAME="E23:37_6"/><b>then </b><i><font color="Green" title="E69">A434</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T1" title="BORSUK_3:th.1">BORSUK_3:1</a></i>;<br/>
<b>thus </b><a NAME="E24:37_6"/>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">WW</font>
 <b>by </b><i><a class="txt" href="#E14:37_6"><i><font color="Green" title="E62">A427</font></i></a>, <a class="txt" href="#E21:37_6"><i><font color="Green" title="E67">A432</font></i></a>, <a class="txt" href="#E22:37_6"><i><font color="Green" title="E68">A433</font></i></a>, <a class="ref" href="borsuk_3.html#T1" title="BORSUK_3:th.1">BORSUK_3:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c21">WW</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">G</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c21">WW</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">V</font> )</span><br/>

<a NAME="E25:37_6"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>;<br/>
<b>hence </b><a NAME="E26:37_6"/>
<font color="Maroon" title="c21">WW</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E20:37_6"><i><font color="Green" title="E66">A431</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c21">WW</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">V</font></span><br/>

<b>let </b><font color="Maroon" title="c22">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">G</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c21">WW</font> or <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font> )</span><br/>

<b>assume </b><a NAME="E27:37_6"/>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">G</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c21">WW</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font></span><br/>

<b>then consider </b><font color="Maroon" title="c23">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E29:37_6"/><i><font color="Green" title="E70">A435</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">WW</font>
 <b>and </b><br/><a NAME="E30:37_6"/><i><font color="Green" title="E71">A436</font></i>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">y0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c25">t0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> , <font color="Maroon" title="c26">N0</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c27">Fn0</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">N0</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <b> such that </b><br/><a NAME="E32:37_6"/><i><font color="Green" title="E72">A437</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c24">y0</font>,<font color="Maroon" title="c25">t0</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E33:37_6"/><i><font color="Green" title="E73">A438</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">Fn0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
 <b>and </b><br/><a NAME="E34:37_6"/><i><font color="Green" title="E74">A439</font></i>: 
<font color="Maroon" title="c24">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">N0</font>
 <b>and </b><br/><a NAME="E35:37_6"/><i><font color="Green" title="E75">A440</font></i>: 
<font color="Maroon" title="c27">Fn0</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E36:37_6"/><i><font color="Green" title="E76">A441</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c26">N0</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c27">Fn0</font>
 <b>and </b><br/><a NAME="E37:37_6"/><i><font color="Green" title="E77">A442</font></i>: 
<font color="Maroon" title="c27">Fn0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c26">N0</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E38:37_6"/>
for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">N0</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c26">N0</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c26">N0</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c27">Fn0</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font>
 <b>by </b><i><a class="txt" href="#E14:37"><i><font color="Green" title="E48">A281</font></i></a></i>;<br/>
<a NAME="E39:37_6"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E29:37_6"><i><font color="Green" title="E70">A435</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E40:37_6"/><b>then </b><i><font color="Green" title="E78">A443</font></i>: 
<font color="Maroon" title="c24">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font>
 
<b>by </b><i><a class="txt" href="#E32:37_6"><i><font color="Green" title="E72">A437</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E41:37_6"/><i><font color="Green" title="E79">A444</font></i>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c24">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E32:37_6"><i><font color="Green" title="E72">A437</font></i></a>, <a class="ref" href="zfmisc_1.html#T128" title="ZFMISC_1:th.128">ZFMISC_1:128</a></i>;<br/>
<b>then </b><font color="Maroon" title="c18">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c24">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">Fn0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c24">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
<b>by </b><i><a class="txt" href="#E18:37"><i><font color="Green" title="E52">A291</font></i></a>, <a class="txt" href="#E7:37_6"><i><font color="Green" title="E58">A423</font></i></a>, <a class="txt" href="#E8:37_6"><i><font color="Green" title="E59">A424</font></i></a>, <a class="txt" href="#E9:37_6"><i><font color="Green" title="E60">A425</font></i></a>, <a class="txt" href="#E32:37_6"><i><font color="Green" title="E72">A437</font></i></a>, <a class="txt" href="#E34:37_6"><i><font color="Green" title="E74">A439</font></i></a>, <a class="txt" href="#E35:37_6"><i><font color="Green" title="E75">A440</font></i></a>, <a class="txt" href="#E36:37_6"><i><font color="Green" title="E76">A441</font></i></a>, <a class="txt" href="#E37:37_6"><i><font color="Green" title="E77">A442</font></i></a>, <a class="txt" href="#E40:37_6"><i><font color="Green" title="E78">A443</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c27">Fn0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
<b>by </b><i><a class="txt" href="#E41:37_6"><i><font color="Green" title="E79">A444</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E43:37_6"/><b>then </b>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Fn</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c19">W</font>
 
<b>by </b><i><a class="txt" href="#E17:37_6"><i><font color="Green" title="E65">A430</font></i></a>, <a class="txt" href="#E21:37_6"><i><font color="Green" title="E67">A432</font></i></a>, <a class="txt" href="#E23:37_6"><i><font color="Green" title="E69">A434</font></i></a>, <a class="txt" href="#E29:37_6"><i><font color="Green" title="E70">A435</font></i></a>, <a class="txt" href="#E30:37_6"><i><font color="Green" title="E71">A436</font></i></a>, <a class="txt" href="#E33:37_6"><i><font color="Green" title="E73">A438</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E44:37_6"/>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">V</font>
 <b>by </b><i><a class="txt" href="#E16:37_6"><i><font color="Green" title="E64">A429</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:37"/>
<font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font> &amp; <font color="Maroon" title="c12">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> &amp; ( for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> holds <br/><font color="Maroon" title="c12">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font> ) )</span><br/>

<b>thus </b><a NAME="E21:37"/>
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">G</font>
 <b>by </b><i><a class="txt" href="#E16:37"><i><font color="Green" title="E50">A283</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> &amp; ( for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> holds <br/><font color="Maroon" title="c12">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font> ) )</span><br/>

<a NAME="E22:37"/><i><font color="Green" title="E53">A445</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E23:37"/><b>then </b><i><font color="Green" title="E54">A446</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">Ft</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">G</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E6:37"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="#E15:37"><i><font color="Green" title="E49">A282</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E24:37"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c10">Ft</font> holds <br/><font color="Maroon" title="c10">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</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="c10">Ft</font> implies <font color="Maroon" title="c10">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<b>assume </b><a NAME="E1:37_7"/><i><font color="Green" title="E55">A447</font></i>: 
<font color="Maroon" title="c13">x</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="c10">Ft</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E2:37_7"/><b>then </b>
<font color="Maroon" title="c13">x</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="c12">G</font>
 
<b>by </b><i><a class="txt" href="#E6:37"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="#E15:37"><i><font color="Green" title="E49">A282</font></i></a>, <a class="txt" href="#E22:37"><i><font color="Green" title="E53">A445</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c15">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> , <font color="Maroon" title="c16">N</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c17">Fn</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <b> such that </b><br/><a NAME="E4:37_7"/><i><font color="Green" title="E56">A448</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c15">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E5:37_7"/><i><font color="Green" title="E57">A449</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <b>and </b><br/><a NAME="E6:37_7"/><i><font color="Green" title="E58">A450</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N</font>
 <b>and </b><br/><a NAME="E7:37_7"/>
( <font color="Maroon" title="c17">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">Fn</font> )
 <b>and </b><br/><a NAME="E8:37_7"/><i><font color="Green" title="E59">A451</font></i>: 
<font color="Maroon" title="c17">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E9:37_7"/>
for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c17">Fn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font>
 <b>by </b><i><a class="txt" href="#E14:37"><i><font color="Green" title="E48">A281</font></i></a></i>;<br/>
<a NAME="E10:37_7"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E4:37_7"><i><font color="Green" title="E56">A448</font></i></a>, <a class="txt" href="#E6:37_7"><i><font color="Green" title="E58">A450</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c10">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c16">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c12">G</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<b>by </b><i><a class="txt" href="#E5:37"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="#E1:37_7"><i><font color="Green" title="E55">A447</font></i></a>, <a class="txt" href="#E5:37_7"><i><font color="Green" title="E57">A449</font></i></a>, <a class="txt" href="#E8:37_7"><i><font color="Green" title="E59">A451</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green" title="E33">Lm14</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:37"/>
<font color="Maroon" title="c12">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font>
 <b>by </b><i><a class="txt" href="#E23:37"><i><font color="Green" title="E54">A446</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> holds <br/><font color="Maroon" title="c12">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font></span><br/>

<b>let </b><font color="Maroon" title="c13">H</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">H</font> &amp; <font color="Maroon" title="c13">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> implies <font color="Maroon" title="c12">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">H</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E26:37"/><i><font color="Green" title="E55">A452</font></i>: 
<font color="Maroon" title="c13">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E27:37"/><i><font color="Green" title="E56">A453</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">H</font>
 <b>and </b><br/><a NAME="E28:37"/><i><font color="Green" title="E57">A454</font></i>: 
<font color="Maroon" title="c13">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">H</font></span><br/>

<a NAME="E29:37"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> holds  <font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font></span><br/>

<b>consider </b><font color="Maroon" title="c15">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c16">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> , <font color="Maroon" title="c17">N</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">Y</font>, <font color="Maroon" title="c18">Fn</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <b> such that </b><br/><a NAME="E2:37_8"/><i><font color="Green" title="E58">A455</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 <b>and </b><br/><a NAME="E3:37_8"/><i><font color="Green" title="E59">A456</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Fn</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>and </b><br/><a NAME="E4:37_8"/><i><font color="Green" title="E60">A457</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">N</font>
 <b>and </b><br/><a NAME="E5:37_8"/>
( <font color="Maroon" title="c18">Fn</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">Fn</font> )
 <b>and </b><br/><a NAME="E6:37_8"/>
<font color="Maroon" title="c18">Fn</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E7:37_8"/><i><font color="Green" title="E61">A458</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c18">Fn</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font>
 <b>by </b><i><a class="txt" href="#E14:37"><i><font color="Green" title="E48">A281</font></i></a></i>;<br/>
<i><font color="Green" title="E62">A459</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E9:37_8"/><b>then </b><i><font color="Green" title="E63">A460</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E2:37_8"><i><font color="Green" title="E58">A455</font></i></a>, <a class="txt" href="#E4:37_8"><i><font color="Green" title="E60">A457</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:37_8"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c8">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:37_8"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">H</font>
 
<b>by </b><i><a class="txt" href="#E4:37"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E12:37_8"/><b>then </b><i><font color="Green" title="E64">A461</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E13:37_8"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">H1</font> = <font color="Maroon" title="c13">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">Y</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">N</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <b>by </b><i><a class="txt" href="#E8:37_8"><i><font color="Green" title="E62">A459</font></i></a>, <a class="txt" href="#E12:37_8"><i><font color="Green" title="E64">A461</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E15:37_8"/><i><font color="Green" title="E65">A462</font></i>: 
<font color="Maroon" title="c19">H1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="#E26:37"><i><font color="Green" title="E55">A452</font></i></a>, <a class="ref" href="topalg_3.html#T18" title="TOPALG_3:th.18">TOPALG_3:18</a></i>;<br/>
<i><font color="Green" title="E66">A463</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c13">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = 
<font color="Maroon" title="c13">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">Y</font>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></i>
<br/>.= 
<font color="Maroon" title="c10">Ft</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E28:37"><i><font color="Green" title="E57">A454</font></i></a>, <a class="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></i>
;<br/>
<a NAME="E17:37_8"/>
<font color="Maroon" title="c9">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E27:37"><i><font color="Green" title="E56">A453</font></i></a>, <a class="ref" href="relat_1.html#T112" title="RELAT_1:th.112">RELAT_1:112</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c12">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c17">N</font>,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> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="txt" href="#E3:37_8"><i><font color="Green" title="E59">A456</font></i></a>, <a class="txt" href="#E7:37_8"><i><font color="Green" title="E61">A458</font></i></a>, <a class="txt" href="#E15:37_8"><i><font color="Green" title="E65">A462</font></i></a>, <a class="txt" href="#E16:37_8"><i><font color="Green" title="E66">A463</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="txt" href="#E8:37_8"><i><font color="Green" title="E62">A459</font></i></a>, <a class="txt" href="#E9:37_8"><i><font color="Green" title="E63">A460</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E30:37"/>
<font color="Maroon" title="c12">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">H</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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