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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Q</font>, <font color="Olive" title="b2">R</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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/>  for <font color="Olive" title="b3">F</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">Q</font></span>)</span><br/>  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b5">P</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>  st <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Olive" title="b3">F</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Olive" title="b3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b3">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b4">i</font>) &amp; <font color="Olive" title="b1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b4">i</font>) holds <br/> ex <font color="Olive" title="b6">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b5">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">R</font></span>)</span> st <br/>( <font color="Olive" title="b6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b5">P</font> &amp; <font color="Olive" title="b6">G</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Q</font>, <font color="Maroon" title="c3">R</font> be   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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span><br/>  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b3">P</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>  st <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Olive" title="b1">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b2">i</font>) &amp; <font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b2">i</font>) holds <br/> ex <font color="Olive" title="b4">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b3">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">R</font></span>)</span> st <br/>( <font color="Olive" title="b4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b4">G</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">P</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>  st <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">i</font>) holds <br/> ex <font color="Olive" title="b3">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b2">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">R</font></span>)</span> st <br/>( <font color="Olive" title="b3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b3">G</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>  st <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">i</font>) &amp; <font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">i</font>) holds <br/> ex <font color="Olive" title="b2">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">R</font></span>)</span> st <br/>( <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b2">G</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">P</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">i</font>) &amp; <font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">i</font>) implies  ex <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">R</font></span>)</span> st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b1">G</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> 
 <span class="kw">and </span><br/><a NAME="E2:10"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E3:10"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">and </span><br/><a NAME="E4:10"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:10"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">and </span><br/><a NAME="E6:10"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">and </span><br/><a NAME="E7:10"/><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">i</font>)
 <span class="kw">and </span><br/><a NAME="E8:10"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
<font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font>
 <span class="kw">and </span><br/><a NAME="E9:10"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
<font color="Maroon" title="c3">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">i</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">R</font></span>)</span> st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b1">G</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">ppi</font>, <font color="Maroon" title="c8">pi1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:10"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
<font color="Maroon" title="c7">ppi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">pi1</font>
 <span class="kw">and </span><br/><a NAME="E12:10"/><span class="lab"><font color="Green" title="E15">A11</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">ppi</font>
 <span class="kw">and </span><br/><a NAME="E13:10"/>
<font color="Maroon" title="c7">ppi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E14:10"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">pi1</font>
 <span class="kw">and </span><br/><a NAME="E15:10"/><span class="lab"><font color="Green" title="E16">A12</font></span>: 
<font color="Maroon" title="c8">pi1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">and </span><br/><a NAME="E16:10"/><span class="lab"><font color="Green" title="E17">A13</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">ppi</font>,<font color="Maroon" title="c8">pi1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E17:10"/>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">ppi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">and </span><br/><a NAME="E18:10"/>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">pi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E6:10"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E8:10"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="jordan5b.html#T7" target="_self" title="JORDAN5B:th.7">Th7</a></span>;<br/>
<a NAME="E19:10"/>
<font color="Maroon" title="c7">ppi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">dd</font> where <font color="Olive" title="b1">dd</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c7">ppi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">dd</font> &amp; <font color="Olive" title="b1">dd</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">pi1</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10"><span class="lab"><font color="Green" title="E14">A10</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">Poz</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">ppi</font>,<font color="Maroon" title="c8">pi1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:10"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="txt" href="#E15:10"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="borsuk_1.html#T40" title="BORSUK_1:th.40">BORSUK_1:40</a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a>, <a class="ref" href="xxreal_1.html#T34" title="XXREAL_1:th.34">XXREAL_1:34</a></span>;<br/>
<span class="lab"><font color="Green" title="E18">A14</font></span>:  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">Poz</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">Poz</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c9">Poz</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<span class="lab"><font color="Green" title="E19">A15</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" title="BORSUK_1:th.40">BORSUK_1:40</a></span>
;<br/>
<a NAME="E23:10"/><span class="lab"><font color="Green" title="E20">A16</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E24:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A17</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Poz</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E22:10"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="ref" href="borsuk_1.html#T40" title="BORSUK_1:th.40">BORSUK_1:40</a>, <a class="ref" href="funct_1.html#T87" title="FUNCT_1:th.87">FUNCT_1:87</a></span>;<br/>
<a NAME="E25:10"/>
<font color="Maroon" title="c9">Poz</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E22:10"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E23:10"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="ref" href="borsuk_1.html#T40" title="BORSUK_1:th.40">BORSUK_1:40</a>, <a class="ref" href="funct_1.html#T87" title="FUNCT_1:th.87">FUNCT_1:87</a></span>;<br/>
<a NAME="E26:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A18</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Poz</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:10"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">gg</font> = <font color="Maroon" title="c4">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">P</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">gg</font> = <font color="Maroon" title="c4">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">P</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">Poz</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:10"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="txt" href="#E26:10"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">SEG</font> =  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">i</font>) as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E9:10"><span class="lab"><font color="Green" title="E13">A9</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E23">A19</font></span>:  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">SEG</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">SEG</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c12">SEG</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E30:10"/><span class="lab"><font color="Green" title="E24">A20</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">gg</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">SEG</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">ii</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">ii</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="c11">gg</font> or <font color="Maroon" title="c13">ii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">SEG</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_4"/>
<font color="Maroon" title="c13">ii</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="c11">gg</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">ii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">SEG</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:10_4"/><span class="lab"><font color="Green" title="E25">A21</font></span>: 
<font color="Maroon" title="c14">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="c11">gg</font>
 <span class="kw">and </span><br/><a NAME="E4:10_4"/><span class="lab"><font color="Green" title="E26">A22</font></span>: 
<font color="Maroon" title="c13">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:10_4"/>
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c9">Poz</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:10"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="#E3:10_4"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<a NAME="E6:10_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E7:10_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c5">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E21:10"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="txt" href="#E3:10_4"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:10_4"/>
<font color="Maroon" title="c13">ii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">SEG</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:10"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="txt" href="#E26:10"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="#E3:10_4"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="txt" href="#E4:10_4"><span class="lab"><font color="Green" title="E26">A22</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c13">SEG</font> = <font color="Maroon" title="c12">SEG</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span> ;<br/>
<a NAME="E32:10"/><span class="lab"><font color="Green" title="E25">A23</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">SEG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:10"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="goboard9.html#T2" title="GOBOARD9:th.2">GOBOARD9:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">hh9</font> = <font color="Maroon" title="c11">gg</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">Poz</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">SEG</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:10"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="txt" href="#E30:10"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>;<br/>
<a NAME="E34:10"/><span class="lab"><font color="Green" title="E26">A24</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E35:10"/><span class="lab"><font color="Green" title="E27">A25</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/>
<a NAME="E36:10"/>
<font color="Maroon" title="c11">gg</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:10"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="#E34:10"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></span>;<br/>
<a NAME="E37:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A26</font></span>: 
<font color="Maroon" title="c14">hh9</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T6" title="TOPMETR:th.6">TOPMETR:6</a></span>;<br/>
<a NAME="E38:10"/><span class="lab"><font color="Green" title="E29">A27</font></span>: 
<font color="Maroon" title="c14">hh9</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:10"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="funct_1.html#T52" title="FUNCT_1:th.52">FUNCT_1:52</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">hh</font> = <font color="Maroon" title="c14">hh9</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">Poz</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">R</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:10"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="goboard9.html#T2" title="GOBOARD9:th.2">GOBOARD9:2</a></span>;<br/>
<a NAME="E40:10"/><span class="lab"><font color="Green" title="E30">A28</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">hh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">Poz</font></span>)</span>
 
<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="E41:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A29</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">hh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Poz</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E32">A30</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c15">hh</font> = 
<font color="Maroon" title="c15">hh</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">hh</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:10"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="ref" href="funct_2.html#T37" title="FUNCT_2:th.37">FUNCT_2:37</a></span>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">SEG</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="txt" href="#E16:10"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E22:10"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E23:10"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E29:10"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="txt" href="#E41:10"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="ref" href="borsuk_1.html#T40" title="BORSUK_1:th.40">BORSUK_1:40</a>, <a class="ref" href="funct_1.html#T87" title="FUNCT_1:th.87">FUNCT_1:87</a>, <a class="ref" href="relat_1.html#T129" title="RELAT_1:th.129">RELAT_1:129</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">A</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c7">ppi</font>,<font color="Maroon" title="c8">pi1</font>) as   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E12:10"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="txt" href="#E15:10"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="topmetr.html#T20" title="TOPMETR:th.20">TOPMETR:20</a>, <a class="ref" href="treal_1.html#T3" title="TREAL_1:th.3">TREAL_1:3</a></span>;<br/>
<a NAME="E44:10"/><span class="lab"><font color="Green" title="E33">A31</font></span>: 
<font color="Maroon" title="c9">Poz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c16">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="topmetr.html#T18" title="TOPMETR:th.18">TOPMETR:18</a></span>;<br/>
<a NAME="E45:10"/>
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c7">ppi</font>,<font color="Maroon" title="c8">pi1</font>) is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="heine.html#T4" title="HEINE:th.4">HEINE:4</a></span>;<br/>
<a NAME="E46:10"/><span class="kw">then </span>
 <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> (<font color="Maroon" title="c7">ppi</font>,<font color="Maroon" title="c8">pi1</font>)</span>)</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="compts_1.html#T1" title="COMPTS_1:th.1">COMPTS_1:1</a></span>;<br/>
<a NAME="E47:10"/><span class="kw">then </span>
 for <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c16">A</font>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Poz</font> holds <br/><font color="Olive" title="b1">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="ref" href="topmetr.html#T18" title="TOPMETR:th.18">TOPMETR:18</a></span>;<br/>
<a NAME="E48:10"/><span class="kw">then </span>
<font color="Maroon" title="c9">Poz</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:10"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="ref" href="compts_1.html#T2" title="COMPTS_1:th.2">COMPTS_1:2</a></span>;<br/>
<a NAME="E49:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A32</font></span>: 
<a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">Poz</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compts_1.html#T3" title="COMPTS_1:th.3">COMPTS_1:3</a></span>;<br/>
<a NAME="E50:10"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">R</font> is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T2" title="TOPMETR:th.2">TOPMETR:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E51:10"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">I[01]</a> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">R</font></span>)</span> st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Olive" title="b1">G</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:10"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="#E32:10"><span class="lab"><font color="Green" title="E25">A23</font></span></a>, <a class="txt" href="#E37:10"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="txt" href="#E38:10"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="txt" href="#E40:10"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="txt" href="#E42:10"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="txt" href="#E49:10"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="ref" href="compts_1.html#T17" title="COMPTS_1:th.17">COMPTS_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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