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<span class="kw">set </span><font color="Maroon" title="c8">L</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">UL</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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> &amp; <font color="Maroon" title="c9">UL</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  implies for <font color="Olive" title="b1">Y</font> being 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><br/> for <font color="Olive" title="b2">F</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Olive" title="b1">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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="b3">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">Y</font>  ex <font color="Olive" title="b4">T</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>  st <br/>( <font color="Olive" title="b4">T</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>  &amp; <font color="Olive" title="b4">T</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="Olive" title="b4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b4">T</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp;  ex <font color="Olive" title="b5">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="Olive" title="b1">Y</font> st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">N</font> &amp; ( for <font color="Olive" title="b6">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="b6">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="Olive" title="b4">T</font> &amp; <font color="Olive" title="b6">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="Olive" title="b4">T</font> holds <br/> ex <font color="Olive" title="b7">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="b7">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> &amp; <font color="Olive" title="b2">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b5">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="Olive" title="b4">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b4">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b6">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="b7">Ui</font> ) ) ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c9">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>
 <span class="kw">and </span><br/><a NAME="E2:36"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c9">UL</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being 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><br/> for <font color="Olive" title="b2">F</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Olive" title="b1">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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="b3">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">Y</font>  ex <font color="Olive" title="b4">T</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>  st <br/>( <font color="Olive" title="b4">T</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>  &amp; <font color="Olive" title="b4">T</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="Olive" title="b4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b4">T</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp;  ex <font color="Olive" title="b5">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="Olive" title="b1">Y</font> st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">N</font> &amp; ( for <font color="Olive" title="b6">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="b6">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="Olive" title="b4">T</font> &amp; <font color="Olive" title="b6">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="Olive" title="b4">T</font> holds <br/> ex <font color="Olive" title="b7">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="b7">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> &amp; <font color="Olive" title="b2">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b5">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="Olive" title="b4">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b6">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b4">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b6">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="b7">Ui</font> ) ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c10">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">Y</font>  ex <font color="Olive" title="b3">T</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>  st <br/>( <font color="Olive" title="b3">T</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>  &amp; <font color="Olive" title="b3">T</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="Olive" title="b3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b3">T</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp;  ex <font color="Olive" title="b4">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="c10">Y</font> st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">N</font> &amp; ( for <font color="Olive" title="b5">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="b5">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="Olive" title="b3">T</font> &amp; <font color="Olive" title="b5">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="Olive" title="b3">T</font> holds <br/> ex <font color="Olive" title="b6">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="b6">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> &amp; <font color="Olive" title="b1">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b4">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="Olive" title="b3">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b3">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b5">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="b6">Ui</font> ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c11">F</font> be   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c10">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c10">Y</font>  ex <font color="Olive" title="b2">T</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>  st <br/>( <font color="Olive" title="b2">T</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>  &amp; <font color="Olive" title="b2">T</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="Olive" title="b2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">T</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp;  ex <font color="Olive" title="b3">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="c10">Y</font> st <br/>( <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; ( for <font color="Olive" title="b4">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="b4">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="Olive" title="b2">T</font> &amp; <font color="Olive" title="b4">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="Olive" title="b2">T</font> holds <br/> ex <font color="Olive" title="b5">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="b5">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>,<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="Olive" title="b2">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b4">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="b5">Ui</font> ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">T</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>  st <br/>( <font color="Olive" title="b1">T</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>  &amp; <font color="Olive" title="b1">T</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="Olive" title="b1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">T</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp;  ex <font color="Olive" title="b2">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="c10">Y</font> st <br/>( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">N</font> &amp; ( for <font color="Olive" title="b3">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="b3">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="Olive" title="b1">T</font> &amp; <font color="Olive" title="b3">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="Olive" title="b1">T</font> holds <br/> ex <font color="Olive" title="b4">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="b4">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>,<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="Olive" title="b1">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b1">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">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="b4">Ui</font> ) ) ) )</span><br/>





<a NAME="E3:36"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <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="c9">UL</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="setfam_1.html#T60" title="SETFAM_1:th.60">SETFAM_1:60</a></span>;<br/>
<a NAME="E4:36"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
for <font color="Olive" title="b1">i</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">U</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 <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="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Olive" title="b1">i</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font> &amp; <font color="Olive" title="b2">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">i</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">U</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 <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="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">i</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c14">U</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:36_1"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
( <font color="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">i</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">U</font> &amp; <font color="Maroon" title="c14">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">U</font> = <font color="Maroon" title="c14">U</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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E2:36_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">U</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">i</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">U</font> &amp; <font color="Maroon" title="c15">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:36_1"/>
( <font color="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">i</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">U</font> &amp; <font color="Maroon" title="c15">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:36"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">U</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 <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="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">UL1</font> = <font color="Maroon" title="c9">UL</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">C</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font>;<br/>
<span class="kw">defpred </span><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> ] <span class="kw">means </span> ex <font color="Olive" title="b1">V</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 <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="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">UL1</font> &amp; <font color="Maroon" title="c11">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="c12">y</font>,$1</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="Olive" title="b1">V</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V</font> );<br/>
<a NAME="E7:36"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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>   ex <font color="Olive" title="b2">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">UL1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">UL1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">i</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<a NAME="E1:36_2"/>
 ex <font color="Olive" title="b1">U</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 <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="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c15">i</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36"><span class="lab"><font color="Green" title="E39">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:36_2"/>
 ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">UL1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">i</font>,<font color="Olive" title="b1">z</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c15">I0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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> ,<font color="Maroon" title="c13">UL1</font><span class="kw"> such that </span><br/><a NAME="E9:36"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c15">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E7:36"><span class="lab"><font color="Green" title="E40">A6</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">M</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="c10">Y</font> ex <font color="Olive" title="b2">O</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">O</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <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="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> );<br/>
<a NAME="E10:36"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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>   ex <font color="Olive" title="b2">z</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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="c10">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> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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="c10">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> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c17">V</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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E2:36_3"/>
<font color="Maroon" title="c17">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">UL1</font>
 <span class="kw">and </span><br/><a NAME="E3:36_3"/><span class="lab"><font color="Green" title="E43">A9</font></span>: 
<font color="Maroon" title="c11">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c16">i</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">V</font>
 <span class="kw">and </span><br/><a NAME="E4:36_3"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<font color="Maroon" title="c15">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E41">A7</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c10">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E6:36_3"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c16">i</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">W</font>
 <span class="kw">and </span><br/><a NAME="E7:36_3"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
<font color="Maroon" title="c18">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E8:36_3"/><span class="lab"><font color="Green" title="E47">A13</font></span>: 
<font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c18">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_3"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">Q</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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c10">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E10:36_3"/><span class="lab"><font color="Green" title="E48">A14</font></span>: 
<font color="Maroon" title="c18">W</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="c19">Q</font>
 <span class="kw">and </span><br/><a NAME="E11:36_3"/><span class="lab"><font color="Green" title="E49">A15</font></span>: 
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="c19">Q</font> holds <br/> ex <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">Y</font> ex <font color="Olive" title="b3">B</font> being  <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">e</font> <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="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <font color="Olive" title="b2">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Olive" title="b3">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36_3"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="ref" href="borsuk_1.html#T45" title="BORSUK_1:th.45">BORSUK_1:45</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:36_3"/><span class="lab"><font color="Green" title="E50">A16</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c16">i</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Z</font>
 <span class="kw">and </span><br/><a NAME="E14:36_3"/><span class="lab"><font color="Green" title="E51">A17</font></span>: 
<font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_3"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="txt" href="#E10:36_3"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">Y</font>, <font color="Maroon" title="c22">B</font> being   <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> <span class="kw"> such that </span><br/><a NAME="E16:36_3"/><span class="lab"><font color="Green" title="E52">A18</font></span>: 
<font color="Maroon" title="c20">Z</font> <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="c21">A</font>,<font color="Maroon" title="c22">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E17:36_3"/><span class="lab"><font color="Green" title="E53">A19</font></span>: 
<font color="Maroon" title="c21">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E18:36_3"/><span class="lab"><font color="Green" title="E54">A20</font></span>: 
<font color="Maroon" title="c22">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_3"><span class="lab"><font color="Green" title="E49">A15</font></span></a>, <a class="txt" href="#E14:36_3"><span class="lab"><font color="Green" title="E51">A17</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">A</font> = <font color="Maroon" title="c21">A</font> as   <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="c10">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:36_3"><span class="lab"><font color="Green" title="E53">A19</font></span></a></span>;<br/>
<a NAME="E20:36_3"/><span class="lab"><font color="Green" title="E55">A21</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36_3"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="#E16:36_3"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">B</font> = <font color="Maroon" title="c22">B</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36_3"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="#E16:36_3"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="txt" href="#E18:36_3"><span class="lab"><font color="Green" title="E54">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">i1</font> = <font color="Maroon" title="c16">i</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="c24">B</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_3"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<a NAME="E23:36_3"/>
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c25">i1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</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="c24">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#T43" title="CONNSP_1:th.43">CONNSP_1:43</a></span>;<br/>
<a NAME="E24:36_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A22</font></span>: 
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c25">i1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#T21" title="CONNSP_2:th.21">CONNSP_2:21</a></span>;<br/>
<a NAME="E25:36_3"/>
 <a href="connsp_3.html#K6" title="CONNSP_3:func.6">Component_of</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c24">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c25">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_3"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="ref" href="connsp_3.html#D7" title="CONNSP_3:def.7">CONNSP_3:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">D</font> =  <a href="connsp_3.html#K6" title="CONNSP_3:func.6">Component_of</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c24">B</font> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_3"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="txt" href="#E24:36_3"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="ref" href="connsp_3.html#T35" title="CONNSP_3:th.35">CONNSP_3:35</a>, <a class="ref" href="tsep_1.html#T17" title="TSEP_1:th.17">TSEP_1:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">z</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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="c10">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c27">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c27">z</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c23">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">O</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">A</font> &amp; <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c27">z</font> <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="c23">A</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c26">D</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">A</font> &amp; <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">D</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c27">z</font> <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="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E28:36_3"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36_3"><span class="lab"><font color="Green" title="E50">A16</font></span></a>, <a class="txt" href="#E16:36_3"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">D</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c27">z</font> <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="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E29:36_3"/>
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_3"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="ref" href="connsp_3.html#T26" title="CONNSP_3:th.26">CONNSP_3:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c27">z</font> <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="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/>

<a NAME="E30:36_3"/>
<font color="Maroon" title="c26">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_3"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="ref" href="topalg_5.html#T3" target="_self" title="TOPALG_5:th.3">Th3</a></span>;<br/>
<a NAME="E31:36_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A23</font></span>: 
<font color="Maroon" title="c27">z</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c24">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></span>;<br/>
<a NAME="E32:36_3"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c24">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:36_3"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="txt" href="#E14:36_3"><span class="lab"><font color="Green" title="E51">A17</font></span></a>, <a class="txt" href="#E16:36_3"><span class="lab"><font color="Green" title="E52">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E33:36_3"/><span class="kw">then </span>
<font color="Maroon" title="c27">z</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c18">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:36_3"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E34:36_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c27">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c18">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:36_3"/>
<font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_3"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="txt" href="#E8:36_3"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">z</font> <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="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/>

<span class="kw">thus </span><a NAME="E36:36_3"/>
<font color="Maroon" title="c27">z</font> <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="c23">A</font>,<font color="Maroon" title="c26">D</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c16">I1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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 class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</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="c10">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></span>)</span><span class="kw"> such that </span><br/><a NAME="E12:36"/><span class="lab"><font color="Green" title="E43">A24</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c16">I1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E10:36"><span class="lab"><font color="Green" title="E42">A8</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c17">C1</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c16">I1</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c10">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<a NAME="E14:36"/><span class="lab"><font color="Green" title="E44">A25</font></span>: 
<font color="Maroon" title="c17">C1</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c10">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">C1</font> or <font color="Maroon" title="c18">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_4"/>
<font color="Maroon" title="c18">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">C1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:36_4"/><span class="lab"><font color="Green" title="E45">A26</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">I1</font>
 <span class="kw">and </span><br/><a NAME="E4:36_4"/><span class="lab"><font color="Green" title="E46">A27</font></span>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:36_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c19">i</font>,<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="txt" href="#E3:36_4"><span class="lab"><font color="Green" title="E45">A26</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:36_4"/>
<font color="Maroon" title="c18">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_4"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:36"/><span class="lab"><font color="Green" title="E45">A28</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">I1</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E16:36"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c17">C1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<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#K4" title="BORSUK_1:func.4">:]</a></span> or <font color="Maroon" title="c18">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="c17">C1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_5"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<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#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">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="c17">C1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">a1</font>, <font color="Maroon" title="c20">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:36_5"/><span class="lab"><font color="Green" title="E46">A29</font></span>: 
<font color="Maroon" title="c19">a1</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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E4:36_5"/><span class="lab"><font color="Green" title="E47">A30</font></span>: 
<font color="Maroon" title="c20">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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 class="kw">and </span><br/><a NAME="E5:36_5"/><span class="lab"><font color="Green" title="E48">A31</font></span>: 
<font color="Maroon" title="c18">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="c19">a1</font>,<font color="Maroon" title="c20">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E6:36_5"/><span class="lab"><font color="Green" title="E49">A32</font></span>: 
<font color="Maroon" title="c19">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_5"><span class="lab"><font color="Green" title="E46">A29</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">a2</font> = <font color="Maroon" title="c20">a2</font> as   <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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_5"><span class="lab"><font color="Green" title="E47">A30</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">M</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="c10">Y</font>, <font color="Maroon" title="c23">O</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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> <span class="kw"> such that </span><br/><a NAME="E9:36_5"/><span class="lab"><font color="Green" title="E50">A33</font></span>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">M</font> &amp; <font color="Maroon" title="c21">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">O</font> )
 <span class="kw">and </span><br/><a NAME="E10:36_5"/>
<font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c22">M</font>,<font color="Maroon" title="c23">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">a2</font>
 <span class="kw">and </span><br/><a NAME="E11:36_5"/><span class="lab"><font color="Green" title="E51">A34</font></span>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">a2</font> <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="c22">M</font>,<font color="Maroon" title="c23">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E43">A24</font></span></a></span>;<br/>
<a NAME="E12:36_5"/>
( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c21">a2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c22">M</font>,<font color="Maroon" title="c23">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c22">M</font>,<font color="Maroon" title="c23">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">C1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36"><span class="lab"><font color="Green" title="E45">A28</font></span></a>, <a class="txt" href="#E9:36_5"><span class="lab"><font color="Green" title="E50">A33</font></span></a>, <a class="txt" href="#E11:36_5"><span class="lab"><font color="Green" title="E51">A34</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:36_5"/>
<font color="Maroon" title="c18">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="c17">C1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_5"><span class="lab"><font color="Green" title="E48">A31</font></span></a>, <a class="txt" href="#E6:36_5"><span class="lab"><font color="Green" title="E49">A32</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A35</font></span>: 
<font color="Maroon" title="c17">C1</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<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#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></span>;<br/>
<a NAME="E18:36"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<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#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_3.html#T27" title="BORSUK_3:th.27">BORSUK_3:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">G</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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c10">Y</font>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E20:36"/><span class="lab"><font color="Green" title="E47">A36</font></span>: 
<font color="Maroon" title="c18">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">C1</font>
 <span class="kw">and </span><br/><a NAME="E21:36"/><span class="lab"><font color="Green" title="E48">A37</font></span>: 
<font color="Maroon" title="c18">G</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<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#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E22:36"/><span class="lab"><font color="Green" title="E49">A38</font></span>: 
<font color="Maroon" title="c18">G</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:36"><span class="lab"><font color="Green" title="E46">A35</font></span></a>, <a class="txt" href="#E14:36"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="ref" href="compts_1.html#D7" title="COMPTS_1:def.7">COMPTS_1:def 7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">NN</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">M</font> where <font color="Olive" title="b1">M</font> is   <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="c10">Y</font> : ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> ) </span>}</span> ;<br/>
<a NAME="E23:36"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">M</font> where <font color="Olive" title="b1">M</font> is   <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="c10">Y</font> : ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">M</font> where <font color="Olive" title="b1">M</font> is   <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="c10">Y</font> : ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> ) </span>}</span>  or <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_6"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">M</font> where <font color="Olive" title="b1">M</font> is   <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="c10">Y</font> : ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font></span><br/>

<a NAME="E2:36_6"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">M</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="c10">Y</font> st <br/>( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:36_6"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">NN</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">M</font> where <font color="Olive" title="b1">M</font> is   <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="c10">Y</font> : ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c10">Y</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">p</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E26:36"/><span class="lab"><font color="Green" title="E50">A39</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">G</font>
 <span class="kw">and </span><br/><a NAME="E27:36"/><span class="lab"><font color="Green" title="E51">A40</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:36"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="ref" href="finset_1.html#D1" title="FINSET_1:def.1">FINSET_1:def 1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">M</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="c10">Y</font> ex <font color="Olive" title="b2">O</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c21">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <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="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> );<br/>
<a NAME="E28:36"/><span class="lab"><font color="Green" title="E52">A41</font></span>: 
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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">p</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">x</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> <font color="Maroon" title="c21">p</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c22">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_7"/>
<font color="Maroon" title="c22">x</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> <font color="Maroon" title="c21">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c22">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<a NAME="E2:36_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A42</font></span>: 
<font color="Maroon" title="c21">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c21">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:36_7"/><span class="lab"><font color="Green" title="E54">A43</font></span>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">I1</font>
 <span class="kw">and </span><br/><a NAME="E5:36_7"/><span class="lab"><font color="Green" title="E55">A44</font></span>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="txt" href="#E26:36"><span class="lab"><font color="Green" title="E50">A39</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">M</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="c10">Y</font>, <font color="Maroon" title="c25">O</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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> <span class="kw"> such that </span><br/><a NAME="E7:36_7"/><span class="lab"><font color="Green" title="E56">A45</font></span>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">M</font> &amp; <font color="Maroon" title="c23">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">O</font> )
 <span class="kw">and </span><br/><a NAME="E8:36_7"/>
<font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c24">M</font>,<font color="Maroon" title="c25">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <span class="kw">and </span><br/><a NAME="E9:36_7"/><span class="lab"><font color="Green" title="E57">A46</font></span>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font> <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="c24">M</font>,<font color="Maroon" title="c25">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="txt" href="#E4:36_7"><span class="lab"><font color="Green" title="E54">A43</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c24">M</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c24">M</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E10:36_7"/>
( <font color="Maroon" title="c24">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c22">x</font>,<font color="Maroon" title="c24">M</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:36"><span class="lab"><font color="Green" title="E50">A39</font></span></a>, <a class="txt" href="#E2:36_7"><span class="lab"><font color="Green" title="E53">A42</font></span></a>, <a class="txt" href="#E5:36_7"><span class="lab"><font color="Green" title="E55">A44</font></span></a>, <a class="txt" href="#E7:36_7"><span class="lab"><font color="Green" title="E56">A45</font></span></a>, <a class="txt" href="#E9:36_7"><span class="lab"><font color="Green" title="E57">A46</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c22">p1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">p</font></span>)</span>,<font color="Maroon" title="c20">NN</font><span class="kw"> such that </span><br/><a NAME="E30:36"/><span class="lab"><font color="Green" title="E53">A47</font></span>: 
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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">p</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c22">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E28:36"><span class="lab"><font color="Green" title="E52">A41</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c23">ICOV</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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">M</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="c10">Y</font> st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> </span>}</span> ;<br/>
<a NAME="E31:36"/><span class="lab"><font color="Green" title="E54">A48</font></span>: 
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<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#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c18">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:36"><span class="lab"><font color="Green" title="E48">A37</font></span></a>, <a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></span>;<br/>
<a NAME="E32:36"/><span class="lab"><font color="Green" title="E55">A49</font></span>: 
<font color="Maroon" title="c12">y</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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E33:36"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<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#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E35:36"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Z</font>
 <span class="kw">and </span><br/><a NAME="E36:36"/><span class="lab"><font color="Green" title="E56">A50</font></span>: 
<font color="Maroon" title="c24">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:36"><span class="lab"><font color="Green" title="E54">A48</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E38:36"/><span class="lab"><font color="Green" title="E57">A51</font></span>: 
<font color="Maroon" title="c25">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">I1</font>
 <span class="kw">and </span><br/><a NAME="E39:36"/><span class="lab"><font color="Green" title="E58">A52</font></span>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="txt" href="#E36:36"><span class="lab"><font color="Green" title="E56">A50</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">M</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="c10">Y</font>, <font color="Maroon" title="c27">O</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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> <span class="kw"> such that </span><br/><a NAME="E41:36"/><span class="lab"><font color="Green" title="E59">A53</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">M</font>
 <span class="kw">and </span><br/><a NAME="E42:36"/>
<font color="Maroon" title="c25">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">O</font>
 <span class="kw">and </span><br/><a NAME="E43:36"/>
<font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c26">M</font>,<font color="Maroon" title="c27">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i</font>
 <span class="kw">and </span><br/><a NAME="E44:36"/><span class="lab"><font color="Green" title="E60">A54</font></span>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i</font> <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="c26">M</font>,<font color="Maroon" title="c27">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="txt" href="#E38:36"><span class="lab"><font color="Green" title="E57">A51</font></span></a></span>;<br/>
<a NAME="E45:36"/><span class="lab"><font color="Green" title="E61">A55</font></span>: 
<font color="Maroon" title="c26">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:36"><span class="lab"><font color="Green" title="E56">A50</font></span></a>, <a class="txt" href="#E39:36"><span class="lab"><font color="Green" title="E58">A52</font></span></a>, <a class="txt" href="#E41:36"><span class="lab"><font color="Green" title="E59">A53</font></span></a>, <a class="txt" href="#E44:36"><span class="lab"><font color="Green" title="E60">A54</font></span></a></span>;<br/>
<a NAME="E46:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A56</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c22">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E47:36"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">NN</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:36_8"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">NN</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">NN</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">p1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c28">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font> or <font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">p1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:36_8"/>
<font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">p1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">M</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="c10">Y</font><span class="kw"> such that </span><br/><a NAME="E4:36_8"/><span class="lab"><font color="Green" title="E63">A57</font></span>: 
<font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">M</font>
 <span class="kw">and </span><br/><a NAME="E5:36_8"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">M</font>
 <span class="kw">and </span><br/><a NAME="E6:36_8"/><span class="lab"><font color="Green" title="E64">A58</font></span>: 
 ex <font color="Olive" title="b1">O</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c29">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c30">O</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw"> such that </span><br/><a NAME="E8:36_8"/><span class="lab"><font color="Green" title="E65">A59</font></span>: 
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c29">M</font>,<font color="Maroon" title="c30">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_8"><span class="lab"><font color="Green" title="E64">A58</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c31">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:36_8"/><span class="lab"><font color="Green" title="E66">A60</font></span>: 
<font color="Maroon" title="c31">b</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> <font color="Maroon" title="c21">p</font>
 <span class="kw">and </span><br/><a NAME="E11:36_8"/><span class="lab"><font color="Green" title="E67">A61</font></span>: 
<font color="Maroon" title="c21">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">b</font> <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="c29">M</font>,<font color="Maroon" title="c30">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:36"><span class="lab"><font color="Green" title="E50">A39</font></span></a>, <a class="txt" href="#E8:36_8"><span class="lab"><font color="Green" title="E65">A59</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c32">M1</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="c10">Y</font>, <font color="Maroon" title="c33">O1</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw"> such that </span><br/><a NAME="E13:36_8"/><span class="lab"><font color="Green" title="E68">A62</font></span>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">M1</font> &amp; <font color="Maroon" title="c21">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">b</font> <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="c32">M1</font>,<font color="Maroon" title="c33">O1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )
 <span class="kw">and </span><br/><a NAME="E14:36_8"/><span class="lab"><font color="Green" title="E69">A63</font></span>: 
<font color="Maroon" title="c22">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">M1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:36"><span class="lab"><font color="Green" title="E53">A47</font></span></a>, <a class="txt" href="#E10:36_8"><span class="lab"><font color="Green" title="E66">A60</font></span></a></span>;<br/>
<a NAME="E15:36_8"/>
<font color="Maroon" title="c32">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_8"><span class="lab"><font color="Green" title="E67">A61</font></span></a>, <a class="txt" href="#E13:36_8"><span class="lab"><font color="Green" title="E68">A62</font></span></a>, <a class="ref" href="zfmisc_1.html#T134" title="ZFMISC_1:th.134">ZFMISC_1:134</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:36_8"/>
<font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:36"><span class="lab"><font color="Green" title="E62">A56</font></span></a>, <a class="txt" href="#E4:36_8"><span class="lab"><font color="Green" title="E63">A57</font></span></a>, <a class="txt" href="#E10:36_8"><span class="lab"><font color="Green" title="E66">A60</font></span></a>, <a class="txt" href="#E14:36_8"><span class="lab"><font color="Green" title="E69">A63</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E48:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A64</font></span>: 
<font color="Maroon" title="c20">NN</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="txt" href="#E46:36"><span class="lab"><font color="Green" title="E62">A56</font></span></a>, <a class="ref" href="finset_1.html#D1" title="FINSET_1:def.1">FINSET_1:def 1</a></span>;<br/>
<a NAME="E49:36"/>
<font color="Maroon" title="c20">NN</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">a</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">Y</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font> or <font color="Maroon" title="c28">a</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_9"/>
<font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c28">a</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E2:36_9"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">M</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="c10">Y</font> st <br/>( <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:36_9"/>
<font color="Maroon" title="c28">a</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">N</font> =  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c20">NN</font> as   <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="c10">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:36"><span class="lab"><font color="Green" title="E63">A64</font></span></a>, <a class="ref" href="tops_2.html#T27" title="TOPS_2:th.27">TOPS_2:27</a></span>;<br/>
<a NAME="E51:36"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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">M</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="c10">Y</font> st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c29">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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">M</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="c10">Y</font> st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> </span>}</span>  or <font color="Maroon" title="c29">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</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/>

<span class="kw">assume </span><a NAME="E1:36_10"/>
<font color="Maroon" title="c29">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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">M</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="c10">Y</font> st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</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/>

<a NAME="E2:36_10"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">O</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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="Maroon" title="c29">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">O</font> &amp;  ex <font color="Olive" title="b2">M</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="c10">Y</font> st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:36_10"/>
<font color="Maroon" title="c29">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">ICOV</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">O</font> where <font color="Olive" title="b1">O</font> is   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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">M</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="c10">Y</font> st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></span>;<br/>
<a NAME="E53:36"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c29">ICOV</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c30">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> or <font color="Maroon" title="c30">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="c29">ICOV</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_11"/>
<font color="Maroon" title="c30">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">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="c29">ICOV</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">a</font> = <font color="Maroon" title="c30">a</font> as   <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></span>;<br/>
<a NAME="E3:36_11"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c31">a</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<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#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:36"><span class="lab"><font color="Green" title="E55">A49</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:36_11"/><span class="lab"><font color="Green" title="E64">A65</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c31">a</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Z</font>
 <span class="kw">and </span><br/><a NAME="E6:36_11"/><span class="lab"><font color="Green" title="E65">A66</font></span>: 
<font color="Maroon" title="c32">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:36"><span class="lab"><font color="Green" title="E54">A48</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c33">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:36_11"/><span class="lab"><font color="Green" title="E66">A67</font></span>: 
<font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">I1</font>
 <span class="kw">and </span><br/><a NAME="E9:36_11"/><span class="lab"><font color="Green" title="E67">A68</font></span>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="txt" href="#E6:36_11"><span class="lab"><font color="Green" title="E65">A66</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c34">M</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="c10">Y</font>, <font color="Maroon" title="c35">O</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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> <span class="kw"> such that </span><br/><a NAME="E11:36_11"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">M</font>
 <span class="kw">and </span><br/><a NAME="E12:36_11"/>
<font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">O</font>
 <span class="kw">and </span><br/><a NAME="E13:36_11"/>
<font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c34">M</font>,<font color="Maroon" title="c35">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font>
 <span class="kw">and </span><br/><a NAME="E14:36_11"/><span class="lab"><font color="Green" title="E68">A69</font></span>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <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="c34">M</font>,<font color="Maroon" title="c35">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="txt" href="#E8:36_11"><span class="lab"><font color="Green" title="E66">A67</font></span></a></span>;<br/>
<a NAME="E15:36_11"/>
( <font color="Maroon" title="c31">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">O</font> &amp; <font color="Maroon" title="c35">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">ICOV</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_11"><span class="lab"><font color="Green" title="E64">A65</font></span></a>, <a class="txt" href="#E6:36_11"><span class="lab"><font color="Green" title="E65">A66</font></span></a>, <a class="txt" href="#E9:36_11"><span class="lab"><font color="Green" title="E67">A68</font></span></a>, <a class="txt" href="#E14:36_11"><span class="lab"><font color="Green" title="E68">A69</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:36_11"/>
<font color="Maroon" title="c30">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="c29">ICOV</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E54:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A70</font></span>: 
<font color="Maroon" title="c29">ICOV</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c30">NCOV</font> being    <a href="rcomp_3.html#M1" title="RCOMP_3:mode.1">IntervalCover</a> of <font color="Maroon" title="c29">ICOV</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c31">T</font> being    <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Maroon" title="c30">NCOV</font>;<br/>
<a NAME="E57:36"/><span class="lab"><font color="Green" title="E65">A71</font></span>: 
<font color="Maroon" title="c29">ICOV</font> is  <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="rcomp_3.html#D1" title="RCOMP_3:def.1">RCOMP_3:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c32">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">ICOV</font> or <font color="Maroon" title="c32">X</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_12"/>
<font color="Maroon" title="c32">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">ICOV</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c32">X</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> </span><br/>

<a NAME="E2:36_12"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">O</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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="Maroon" title="c32">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">O</font> &amp;  ex <font color="Olive" title="b2">M</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="c10">Y</font> st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:36_12"/>
<font color="Maroon" title="c32">X</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E58:36"/><span class="lab"><font color="Green" title="E66">A72</font></span>: 
<font color="Maroon" title="c29">ICOV</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">a</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c32">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">ICOV</font> or <font color="Maroon" title="c32">a</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_13"/>
<font color="Maroon" title="c32">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">ICOV</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c32">a</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<a NAME="E2:36_13"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">O</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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="Maroon" title="c32">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">O</font> &amp;  ex <font color="Olive" title="b2">M</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="c10">Y</font> st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">M</font>,<font color="Olive" title="b1">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:36_13"/>
<font color="Maroon" title="c32">a</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">T</font> = <font color="Maroon" title="c31">T</font> as  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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:36"><span class="lab"><font color="Green" title="E64">A70</font></span></a>, <a class="txt" href="#E57:36"><span class="lab"><font color="Green" title="E65">A71</font></span></a>, <a class="txt" href="topalg_5.html#E40"><span class="lab"><font color="Green" title="E34">Lm15</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c32">T</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c32">T</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>  &amp; <font color="Maroon" title="c32">T</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="c32">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c32">T</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp;  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="c10">Y</font> st <br/>( <font color="Maroon" title="c12">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="c32">T</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="c32">T</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="c9">UL</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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="c32">T</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="c32">T</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> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E60:36"/>
( <font color="Maroon" title="c32">T</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>  &amp; <font color="Maroon" title="c32">T</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="c32">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:36"><span class="lab"><font color="Green" title="E64">A70</font></span></a>, <a class="txt" href="#E58:36"><span class="lab"><font color="Green" title="E66">A72</font></span></a>, <a class="txt" href="#E57:36"><span class="lab"><font color="Green" title="E65">A71</font></span></a>, <a class="ref" href="rcomp_3.html#D3" title="RCOMP_3:def.3">RCOMP_3:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c32">T</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp;  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="c10">Y</font> st <br/>( <font color="Maroon" title="c12">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="c32">T</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="c32">T</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="c9">UL</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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="c32">T</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="c32">T</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> ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E61:36"/>
<font color="Maroon" title="c32">T</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:36"><span class="lab"><font color="Green" title="E64">A70</font></span></a>, <a class="txt" href="#E58:36"><span class="lab"><font color="Green" title="E66">A72</font></span></a>, <a class="txt" href="#E57:36"><span class="lab"><font color="Green" title="E65">A71</font></span></a>, <a class="ref" href="rcomp_3.html#T79" title="RCOMP_3:th.79">RCOMP_3:79</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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="c10">Y</font> st <br/>( <font color="Maroon" title="c12">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="c32">T</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="c32">T</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="c9">UL</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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="c32">T</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="c32">T</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> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c28">N</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">N</font> &amp; ( 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="c32">T</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="c32">T</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="c9">UL</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c28">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="c32">T</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="c32">T</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> ) ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_14"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c33">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font> implies <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Z</font> )</span><br/><span class="kw">assume </span><a NAME="E1:36_14"/>
<font color="Maroon" title="c33">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Z</font></span><br/><a NAME="E2:36_14"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">M</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="c10">Y</font> st <br/>( <font color="Maroon" title="c33">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp;  ex <font color="Olive" title="b2">O</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:36_14"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E63:36"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:36"><span class="lab"><font color="Green" title="E61">A55</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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="c32">T</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="c32">T</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="c9">UL</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c28">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="c32">T</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="c32">T</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> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c33">i</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">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="c32">T</font> &amp; <font color="Maroon" title="c33">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="c32">T</font> implies  ex <font color="Olive" title="b1">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="b1">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c28">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="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">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="b1">Ui</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E64:36"/><span class="lab"><font color="Green" title="E67">A73</font></span>: 
<font color="Maroon" title="c33">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="c32">T</font>
 <span class="kw">and </span><br/><a NAME="E65:36"/><span class="lab"><font color="Green" title="E68">A74</font></span>: 
<font color="Maroon" title="c33">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="c32">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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="b1">Ui</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c28">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="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">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="b1">Ui</font> )</span><br/>

<a NAME="E66:36"/><span class="lab"><font color="Green" title="E69">A75</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:36"><span class="lab"><font color="Green" title="E67">A73</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E67:36"/><span class="lab"><font color="Green" title="E70">A76</font></span>: 
<font color="Maroon" title="c33">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c32">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:36"><span class="lab"><font color="Green" title="E68">A74</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E68:36"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c30">NCOV</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:36"><span class="lab"><font color="Green" title="E64">A70</font></span></a>, <a class="txt" href="#E58:36"><span class="lab"><font color="Green" title="E66">A72</font></span></a>, <a class="txt" href="#E57:36"><span class="lab"><font color="Green" title="E65">A71</font></span></a>, <a class="ref" href="rcomp_3.html#D3" title="RCOMP_3:def.3">RCOMP_3:def 3</a></span>;<br/>
<a NAME="E69:36"/><span class="kw">then </span>
<font color="Maroon" title="c33">i</font> <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="c30">NCOV</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:36"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E70:36"/><span class="kw">then </span>
<font color="Maroon" title="c33">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="c30">NCOV</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:36"><span class="lab"><font color="Green" title="E69">A75</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E71:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A77</font></span>: 
<font color="Maroon" title="c30">NCOV</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c30">NCOV</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E72:36"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c30">NCOV</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">ICOV</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:36"><span class="lab"><font color="Green" title="E64">A70</font></span></a>, <a class="txt" href="#E58:36"><span class="lab"><font color="Green" title="E66">A72</font></span></a>, <a class="txt" href="#E57:36"><span class="lab"><font color="Green" title="E65">A71</font></span></a>, <a class="ref" href="rcomp_3.html#D2" title="RCOMP_3:def.2">RCOMP_3:def 2</a></span>;<br/>
<a NAME="E73:36"/><span class="kw">then </span>
<font color="Maroon" title="c30">NCOV</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">ICOV</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:36"><span class="lab"><font color="Green" title="E71">A77</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c34">O</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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> <span class="kw"> such that </span><br/><a NAME="E75:36"/><span class="lab"><font color="Green" title="E72">A78</font></span>: 
<font color="Maroon" title="c30">NCOV</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">O</font>
 <span class="kw">and </span><br/><a NAME="E76:36"/><span class="lab"><font color="Green" title="E73">A79</font></span>: 
 ex <font color="Olive" title="b1">M</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="c10">Y</font> st <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">M</font>,<font color="Maroon" title="c34">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c35">M</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="c10">Y</font><span class="kw"> such that </span><br/><a NAME="E78:36"/><span class="lab"><font color="Green" title="E74">A80</font></span>: 
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c35">M</font>,<font color="Maroon" title="c34">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:36"><span class="lab"><font color="Green" title="E73">A79</font></span></a></span>;<br/>
<a NAME="E79:36"/>
<font color="Maroon" title="c33">i</font> <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="c32">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:36"><span class="lab"><font color="Green" title="E70">A76</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E80:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A81</font></span>: 
<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="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c33">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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c34">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:36"><span class="lab"><font color="Green" title="E64">A70</font></span></a>, <a class="txt" href="#E58:36"><span class="lab"><font color="Green" title="E66">A72</font></span></a>, <a class="txt" href="#E57:36"><span class="lab"><font color="Green" title="E65">A71</font></span></a>, <a class="txt" href="#E66:36"><span class="lab"><font color="Green" title="E69">A75</font></span></a>, <a class="txt" href="#E75:36"><span class="lab"><font color="Green" title="E72">A78</font></span></a>, <a class="ref" href="rcomp_3.html#T80" title="RCOMP_3:th.80">RCOMP_3:80</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c36">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E82:36"/><span class="lab"><font color="Green" title="E76">A82</font></span>: 
<font color="Maroon" title="c36">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">I1</font>
 <span class="kw">and </span><br/><a NAME="E83:36"/><span class="lab"><font color="Green" title="E77">A83</font></span>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> <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="c35">M</font>,<font color="Maroon" title="c34">O</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36"><span class="lab"><font color="Green" title="E47">A36</font></span></a>, <a class="txt" href="#E78:36"><span class="lab"><font color="Green" title="E74">A80</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c37">V</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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E85:36"/><span class="lab"><font color="Green" title="E78">A84</font></span>: 
<font color="Maroon" title="c37">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">UL1</font>
 <span class="kw">and </span><br/><a NAME="E86:36"/><span class="lab"><font color="Green" title="E79">A85</font></span>: 
<font color="Maroon" title="c11">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="c12">y</font>,<font color="Maroon" title="c36">j</font></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">V</font>
 <span class="kw">and </span><br/><a NAME="E87:36"/><span class="lab"><font color="Green" title="E80">A86</font></span>: 
<font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="txt" href="#E82:36"><span class="lab"><font color="Green" title="E76">A82</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c38">V</font> = <font color="Maroon" title="c37">V</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 <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:36"><span class="lab"><font color="Green" title="E79">A85</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c38">V</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c38">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font> &amp; <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c28">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="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">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="Maroon" title="c38">V</font> )</span><br/>

<span class="kw">thus </span><a NAME="E89:36"/>
<font color="Maroon" title="c38">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">UL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E85:36"><span class="lab"><font color="Green" title="E78">A84</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c28">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="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">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="Maroon" title="c38">V</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c39">M1</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="c10">Y</font>, <font color="Maroon" title="c40">O1</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</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> <span class="kw"> such that </span><br/><a NAME="E91:36"/><span class="lab"><font color="Green" title="E81">A87</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c39">M1</font>
 <span class="kw">and </span><br/><a NAME="E92:36"/><span class="lab"><font color="Green" title="E82">A88</font></span>: 
<font color="Maroon" title="c36">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c40">O1</font>
 <span class="kw">and </span><br/><a NAME="E93:36"/><span class="lab"><font color="Green" title="E83">A89</font></span>: 
<font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c39">M1</font>,<font color="Maroon" title="c40">O1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">I0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font>
 <span class="kw">and </span><br/><a NAME="E94:36"/><span class="lab"><font color="Green" title="E84">A90</font></span>: 
<font color="Maroon" title="c16">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">j</font> <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="c39">M1</font>,<font color="Maroon" title="c40">O1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="txt" href="#E82:36"><span class="lab"><font color="Green" title="E76">A82</font></span></a></span>;<br/>
<a NAME="E95:36"/>
<font color="Maroon" title="c35">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c39">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:36"><span class="lab"><font color="Green" title="E77">A83</font></span></a>, <a class="txt" href="#E91:36"><span class="lab"><font color="Green" title="E81">A87</font></span></a>, <a class="txt" href="#E92:36"><span class="lab"><font color="Green" title="E82">A88</font></span></a>, <a class="txt" href="#E94:36"><span class="lab"><font color="Green" title="E84">A90</font></span></a>, <a class="ref" href="zfmisc_1.html#T134" title="ZFMISC_1:th.134">ZFMISC_1:134</a></span>;<br/>
<a NAME="E96:36"/><span class="kw">then </span>
<font color="Maroon" title="c35">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">NN</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:36"><span class="lab"><font color="Green" title="E74">A80</font></span></a>, <a class="txt" href="#E91:36"><span class="lab"><font color="Green" title="E81">A87</font></span></a></span>;<br/>
<a NAME="E97:36"/><span class="kw">then </span>
<font color="Maroon" title="c28">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c35">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></span>;<br/>
<a NAME="E98:36"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c28">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="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">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> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c39">M1</font>,<font color="Maroon" title="c40">O1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:36"><span class="lab"><font color="Green" title="E77">A83</font></span></a>, <a class="txt" href="#E94:36"><span class="lab"><font color="Green" title="E84">A90</font></span></a>, <a class="txt" href="#E80:36"><span class="lab"><font color="Green" title="E75">A81</font></span></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></span>;<br/>
<a NAME="E99:36"/><span class="kw">then </span>
<font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c28">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="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">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="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c39">M1</font>,<font color="Maroon" title="c40">O1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></span>;<br/>
<span class="kw">hence </span><a NAME="E100:36"/>
<font color="Maroon" title="c11">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c28">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="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c33">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c32">T</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c33">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="Maroon" title="c38">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E93:36"><span class="lab"><font color="Green" title="E83">A89</font></span></a>, <a class="txt" href="#E87:36"><span class="lab"><font color="Green" title="E80">A86</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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