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<span class="kw">let </span><font color="Maroon" title="c8">x0</font>, <font color="Maroon" title="c9">y0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c9">y0</font><br/> for <font color="Olive" title="b3">F</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font><br/> for <font color="Olive" title="b4">xt</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Olive" title="b4">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">x0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/> ex <font color="Olive" title="b5">yt</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  ex <font color="Olive" title="b6">Pt</font>, <font color="Olive" title="b7">Qt</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b4">xt</font>,<font color="Olive" title="b5">yt</font> ex <font color="Olive" title="b8">Ft</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b6">Pt</font>,<font color="Olive" title="b7">Qt</font> st <br/>( <font color="Olive" title="b6">Pt</font>,<font color="Olive" title="b7">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Olive" title="b3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b8">Ft</font> &amp; <font color="Olive" title="b5">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b9">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b6">Pt</font>,<font color="Olive" title="b7">Qt</font>  st <font color="Olive" title="b3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b9">F1</font> holds <br/><font color="Olive" title="b8">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b9">F1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">P</font>, <font color="Maroon" title="c11">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c9">y0</font>; <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="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font><br/> for <font color="Olive" title="b2">xt</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Olive" title="b2">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">x0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/> ex <font color="Olive" title="b3">yt</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  ex <font color="Olive" title="b4">Pt</font>, <font color="Olive" title="b5">Qt</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b2">xt</font>,<font color="Olive" title="b3">yt</font> ex <font color="Olive" title="b6">Ft</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b4">Pt</font>,<font color="Olive" title="b5">Qt</font> st <br/>( <font color="Olive" title="b4">Pt</font>,<font color="Olive" title="b5">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Olive" title="b1">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b6">Ft</font> &amp; <font color="Olive" title="b3">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b7">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b4">Pt</font>,<font color="Olive" title="b5">Qt</font>  st <font color="Olive" title="b1">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b7">F1</font> holds <br/><font color="Olive" title="b6">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b7">F1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">F</font> be    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font>; <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">xt</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Olive" title="b1">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">x0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/> ex <font color="Olive" title="b2">yt</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  ex <font color="Olive" title="b3">Pt</font>, <font color="Olive" title="b4">Qt</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">xt</font>,<font color="Olive" title="b2">yt</font> ex <font color="Olive" title="b5">Ft</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b3">Pt</font>,<font color="Olive" title="b4">Qt</font> st <br/>( <font color="Olive" title="b3">Pt</font>,<font color="Olive" title="b4">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">Ft</font> &amp; <font color="Olive" title="b2">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b6">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b3">Pt</font>,<font color="Olive" title="b4">Qt</font>  st <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b6">F1</font> holds <br/><font color="Olive" title="b5">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b6">F1</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c13">xt</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <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="c10">P</font>,<font color="Maroon" title="c11">Q</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Maroon" title="c13">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">x0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies  ex <font color="Olive" title="b1">yt</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  ex <font color="Olive" title="b2">Pt</font>, <font color="Olive" title="b3">Qt</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">xt</font>,<font color="Olive" title="b1">yt</font> ex <font color="Olive" title="b4">Ft</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b2">Pt</font>,<font color="Olive" title="b3">Qt</font> st <br/>( <font color="Olive" title="b2">Pt</font>,<font color="Olive" title="b3">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">Ft</font> &amp; <font color="Olive" title="b1">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b5">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b2">Pt</font>,<font color="Olive" title="b3">Qt</font>  st <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">F1</font> holds <br/><font color="Olive" title="b4">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b5">F1</font> ) ) )</span><br/>







<span class="kw">consider </span><font color="Maroon" title="c14">cP1</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c8">x0</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">g1</font> = <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c13">xt</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">cP2</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c9">y0</font>;<br/>
<span class="kw">assume </span><a NAME="E3:39"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c10">P</font>,<font color="Maroon" title="c11">Q</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</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="c13">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">x0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  ex <font color="Olive" title="b1">yt</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  ex <font color="Olive" title="b2">Pt</font>, <font color="Olive" title="b3">Qt</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">xt</font>,<font color="Olive" title="b1">yt</font> ex <font color="Olive" title="b4">Ft</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b2">Pt</font>,<font color="Olive" title="b3">Qt</font> st <br/>( <font color="Olive" title="b2">Pt</font>,<font color="Olive" title="b3">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">Ft</font> &amp; <font color="Olive" title="b1">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b5">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b2">Pt</font>,<font color="Olive" title="b3">Qt</font>  st <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">F1</font> holds <br/><font color="Olive" title="b4">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b5">F1</font> ) ) )</span><br/>

<a NAME="E4:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c12">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="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:39"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c13">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">x0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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">yt</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  ex <font color="Olive" title="b2">Pt</font>, <font color="Olive" title="b3">Qt</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">xt</font>,<font color="Olive" title="b1">yt</font> ex <font color="Olive" title="b4">Ft</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b2">Pt</font>,<font color="Olive" title="b3">Qt</font> st <br/>( <font color="Olive" title="b2">Pt</font>,<font color="Olive" title="b3">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">Ft</font> &amp; <font color="Olive" title="b1">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b5">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b2">Pt</font>,<font color="Olive" title="b3">Qt</font>  st <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b5">F1</font> holds <br/><font color="Olive" title="b4">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b5">F1</font> ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">ft</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <span class="kw"> such that </span><br/><a NAME="E7:39"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c17">ft</font> <a href="seq_1.html#K1" title="SEQ_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="c13">xt</font>
 <span class="kw">and </span><br/><a NAME="E8:39"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<font color="Maroon" title="c10">P</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">ft</font>
 <span class="kw">and </span><br/><a NAME="E9:39"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<font color="Maroon" title="c17">ft</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E10:39"/>
for <font color="Olive" title="b1">f1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c10">P</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Olive" title="b1">f1</font> <a href="seq_1.html#K1" title="SEQ_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="c13">xt</font> holds <br/><font color="Maroon" title="c17">ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#T23" target="_self" title="TOPALG_5:th.23">Th23</a></span>;<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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1;<br/>
<a NAME="E11:39"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
for <font color="Olive" title="b1">x</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> <br/> for <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>  ex <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">Ft</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E13:39"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
for <font color="Olive" title="b1">y</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> <br/> for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c18">Ft</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E45">A7</font></span></a>);<br/></span>
<a NAME="E14:39"/>
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">xt</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="c8">x0</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="txt" href="#E5:39"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></span>;<br/>
<a NAME="E15:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">xt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x0</font>
 
<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="E16:39"/><span class="lab"><font color="Green" title="E48">A10</font></span>: 
for <font color="Olive" title="b1">x</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>  holds  <font color="Maroon" title="c14">cP1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c13">xt</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</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"> <font color="Maroon" title="c14">cP1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c13">xt</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c14">cP1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font> = 
<font color="Maroon" title="c8">x0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_3.html#T22" title="TOPALG_3:th.22">TOPALG_3:22</a></span>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c13">xt</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c13">xt</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</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="c19">ft1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <span class="kw"> such that </span><br/><a NAME="E18:39"/>
<font color="Maroon" title="c19">ft1</font> <a href="seq_1.html#K1" title="SEQ_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="c13">xt</font>
 <span class="kw">and </span><br/><a NAME="E19:39"/>
<font color="Maroon" title="c14">cP1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">ft1</font>
 <span class="kw">and </span><br/><a NAME="E20:39"/>
<font color="Maroon" title="c19">ft1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E21:39"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
for <font color="Olive" title="b1">f1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c14">cP1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Olive" title="b1">f1</font> <a href="seq_1.html#K1" title="SEQ_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="c13">xt</font> holds <br/><font color="Maroon" title="c19">ft1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="topalg_5.html#T23" target="_self" title="TOPALG_5:th.23">Th23</a></span>;<br/>
<a NAME="E22:39"/>
<span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c13">xt</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xt</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></span>;<br/>
<a NAME="E23:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A12</font></span>: 
<font color="Maroon" title="c19">ft1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c13">xt</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:39"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E16:39"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/>
<a NAME="E24:39"/><span class="lab"><font color="Green" title="E51">A13</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">Ft</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<a NAME="E25:39"/><span class="lab"><font color="Green" title="E52">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">Ft</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<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="E26:39"/><span class="lab"><font color="Green" title="E53">A15</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span></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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">Ft</font> = <font color="Maroon" title="c18">Ft</font> as   <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"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="topalg_5.html#E39"><span class="lab"><font color="Green" title="E33">Lm14</font></span></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></span>;<br/>
<a NAME="E28:39"/><span class="lab"><font color="Green" title="E54">A16</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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">Ft</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">Ft</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">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="c21">x</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> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">Ft</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">Ft</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_2"/><span class="lab"><font color="Green" title="E55">A17</font></span>: 
<font color="Maroon" title="c21">x</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> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">Ft</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">Ft</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c22">x1</font>, <font color="Maroon" title="c23">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_2"/><span class="lab"><font color="Green" title="E56">A18</font></span>: 
<font color="Maroon" title="c22">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <span class="kw">and </span><br/><a NAME="E4:39_2"/><span class="lab"><font color="Green" title="E57">A19</font></span>: 
<font color="Maroon" title="c23">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:39_2"/><span class="lab"><font color="Green" title="E58">A20</font></span>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">x1</font>,<font color="Maroon" title="c23">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:39"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="topalg_5.html#E39"><span class="lab"><font color="Green" title="E33">Lm14</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E6:39_2"/>
<font color="Maroon" title="c23">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_2"><span class="lab"><font color="Green" title="E57">A19</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> = 
<font color="Maroon" title="c12">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">x1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:39"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E5:39_2"><span class="lab"><font color="Green" title="E58">A20</font></span></a>, <a class="txt" href="topalg_5.html#E39"><span class="lab"><font color="Green" title="E33">Lm14</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c17">ft</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E3:39_2"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></span>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c22">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_2"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c22">x1</font>,<font color="Maroon" title="c23">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E3:39_2"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E4:39_2"><span class="lab"><font color="Green" title="E57">A19</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">Ft</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E5:39_2"><span class="lab"><font color="Green" title="E58">A20</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
;<br/> <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="E29:39"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><br/> for <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c20">Ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c20">Ft</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Maroon" title="c20">Ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> ex <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c20">Ft</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c22">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">Ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">V</font> &amp; <font color="Maroon" title="c22">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c20">Ft</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">V</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:39_3"/><span class="lab"><font color="Green" title="E55">A21</font></span>: 
( <font color="Maroon" title="c20">Ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">V</font> &amp; <font color="Maroon" title="c22">V</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">  ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c20">Ft</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">V</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c23">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> , <font color="Maroon" title="c24">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E3:39_3"/><span class="lab"><font color="Green" title="E56">A22</font></span>: 
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c24">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:39"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></span>;<br/>
<a NAME="E4:39_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c24">p2</font>,<font color="Maroon" title="c20">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c23">p1</font>,<font color="Maroon" title="c24">p2</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="topalg_5.html#E39"><span class="lab"><font color="Green" title="E33">Lm14</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">W1</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="E6:39_3"/><span class="lab"><font color="Green" title="E57">A23</font></span>: 
<font color="Maroon" title="c23">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">W1</font>
 <span class="kw">and </span><br/><a NAME="E7:39_3"/><span class="lab"><font color="Green" title="E58">A24</font></span>: 
<font color="Maroon" title="c25">W1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E8:39_3"/><span class="lab"><font color="Green" title="E59">A25</font></span>: 
<font color="Maroon" title="c17">ft</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c25">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E1:39_3"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="txt" href="#E3:39_3"><span class="lab"><font color="Green" title="E56">A22</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">reconsider </span><font color="Maroon" title="c26">W1</font> = <font color="Maroon" title="c25">W1</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_3"><span class="lab"><font color="Green" title="E57">A23</font></span></a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c27">W</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c26">W1</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span></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="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">W</font> &amp; <font color="Maroon" title="c27">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c20">Ft</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c27">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">V</font> )</span><br/>

<span class="kw">thus </span><a NAME="E10:39_3"/>
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_3"><span class="lab"><font color="Green" title="E56">A22</font></span></a>, <a class="txt" href="#E6:39_3"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</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">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c20">Ft</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c27">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">V</font> )</span><br/>

<span class="kw">thus </span><a NAME="E11:39_3"/>
<font color="Maroon" title="c27">W</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="#E7:39_3"><span class="lab"><font color="Green" title="E58">A24</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"> <font color="Maroon" title="c20">Ft</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c27">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c28">y</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Ft</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c27">W</font> or <font color="Maroon" title="c28">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E12:39_3"/>
<font color="Maroon" title="c28">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Ft</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c27">W</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E14:39_3"/><span class="lab"><font color="Green" title="E60">A26</font></span>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">W</font>
 <span class="kw">and </span><br/><a NAME="E15:39_3"/><span class="lab"><font color="Green" title="E61">A27</font></span>: 
<font color="Maroon" title="c28">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c30">x1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c26">W1</font>, <font color="Maroon" title="c31">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="tex_2.html#K1" title="TEX_2:func.1">Sspace</a> <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">0[01]</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E17:39_3"/><span class="lab"><font color="Green" title="E62">A28</font></span>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c30">x1</font>,<font color="Maroon" title="c31">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:39_3"><span class="lab"><font color="Green" title="E60">A26</font></span></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></span>;<br/>
<a NAME="E18:39_3"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c30">x1</font>,<font color="Maroon" title="c31">x2</font>,<font color="Maroon" title="c20">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c30">x1</font>,<font color="Maroon" title="c31">x2</font>] &amp; <font color="Maroon" title="c17">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">ft</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c26">W1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="topalg_5.html#E39"><span class="lab"><font color="Green" title="E33">Lm14</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:39_3"/>
<font color="Maroon" title="c28">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_3"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="txt" href="#E15:39_3"><span class="lab"><font color="Green" title="E61">A27</font></span></a>, <a class="txt" href="#E17:39_3"><span class="lab"><font color="Green" title="E62">A28</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="E30:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A29</font></span>: 
<font color="Maroon" title="c20">Ft</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="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c21">yt</font> = <font color="Maroon" title="c17">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</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">Pt</font>, <font color="Olive" title="b2">Qt</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">xt</font>,<font color="Maroon" title="c21">yt</font> ex <font color="Olive" title="b3">Ft</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b1">Pt</font>,<font color="Olive" title="b2">Qt</font> st <br/>( <font color="Olive" title="b1">Pt</font>,<font color="Olive" title="b2">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">Ft</font> &amp; <font color="Maroon" title="c21">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b4">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Olive" title="b1">Pt</font>,<font color="Olive" title="b2">Qt</font>  st <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">F1</font> holds <br/><font color="Olive" title="b3">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">F1</font> ) )</span><br/>

<a NAME="E31:39"/><span class="lab"><font color="Green" title="E56">A30</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c4">j0</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="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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="topalg_5.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</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="c22">ft</font> = <font color="Maroon" title="c17">ft</font> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">xt</font>,<font color="Maroon" title="c21">yt</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></span>;<br/>
<a NAME="E33:39"/><span class="lab"><font color="Green" title="E57">A31</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c4">j0</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="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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="topalg_5.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E34:39"/><span class="lab"><font color="Green" title="E58">A32</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E35:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A33</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="topalg_5.html#E3"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a>, <a class="txt" href="topalg_5.html#E5"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></span>;<br/>
<a NAME="E36:39"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">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="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<a NAME="E37:39"/><span class="kw">then </span>
<font color="Maroon" title="c12">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c20">Ft</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:39"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E24:39"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E28:39"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="topalg_5.html#E35"><span class="lab"><font color="Green" title="E29">Lm12</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<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>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <span class="kw"> such that </span><br/><a NAME="E39:39"/><span class="lab"><font color="Green" title="E60">A34</font></span>: 
<font color="Maroon" title="c23">G</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E40:39"/><span class="lab"><font color="Green" title="E61">A35</font></span>: 
<font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">G</font>
 <span class="kw">and </span><br/><a NAME="E41:39"/><span class="lab"><font color="Green" title="E62">A36</font></span>: 
<font color="Maroon" title="c23">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Ft</font>
 <span class="kw">and </span><br/><a NAME="E42:39"/><span class="lab"><font color="Green" title="E63">A37</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<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>,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">H</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b1">H</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Ft</font> holds <br/><font color="Maroon" title="c23">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E30:39"><span class="lab"><font color="Green" title="E55">A29</font></span></a>, <a class="ref" href="topalg_5.html#T22" target="_self" title="TOPALG_5:th.22">Th22</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">sM0</font> =  <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c23">G</font>;<br/>
<a NAME="E43:39"/><span class="lab"><font color="Green" title="E64">A38</font></span>: 
for <font color="Olive" title="b1">x</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>  holds  <font color="Maroon" title="c14">cP1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c23">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x</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"> <font color="Maroon" title="c14">cP1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c23">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c23">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font> = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c23">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c25">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">G</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c25">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="topalg_5.html#E5"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a>, <a class="ref" href="binop_1.html#T30" title="BINOP_1:th.30">BINOP_1:30</a></span>
<br/>.= 
<font color="Maroon" title="c8">x0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E40:39"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c14">cP1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_3.html#T22" title="TOPALG_3:th.22">TOPALG_3:22</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">set </span><font color="Maroon" title="c25">g2</font> = <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c21">yt</font>;<br/>
<span class="lab"><font color="Green" title="E65">A39</font></span>: <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">yt</font> = 
<font color="Maroon" title="c10">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<font color="Maroon" title="c9">y0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></span>
;<br/>
<a NAME="E45:39"/><span class="lab"><font color="Green" title="E66">A40</font></span>: 
for <font color="Olive" title="b1">x</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>  holds  <font color="Maroon" title="c16">cP2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c21">yt</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">x</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"> <font color="Maroon" title="c16">cP2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c21">yt</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c16">cP2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x</font> = 
<font color="Maroon" title="c9">y0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_3.html#T22" title="TOPALG_3:th.22">TOPALG_3:22</a></span>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c21">yt</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:39"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c21">yt</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</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="E46:39"/><span class="lab"><font color="Green" title="E67">A41</font></span>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">yt</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="c9">y0</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="txt" href="#E44:39"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E47:39"/><span class="kw">then </span>
<font color="Maroon" title="c21">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</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="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">ft2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a> <span class="kw"> such that </span><br/><a NAME="E49:39"/>
<font color="Maroon" title="c26">ft2</font> <a href="seq_1.html#K1" title="SEQ_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="c21">yt</font>
 <span class="kw">and </span><br/><a NAME="E50:39"/>
<font color="Maroon" title="c16">cP2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c26">ft2</font>
 <span class="kw">and </span><br/><a NAME="E51:39"/>
<font color="Maroon" title="c26">ft2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E52:39"/><span class="lab"><font color="Green" title="E68">A42</font></span>: 
for <font color="Olive" title="b1">f1</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>   st <font color="Olive" title="b1">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c16">cP2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Olive" title="b1">f1</font> <a href="seq_1.html#K1" title="SEQ_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="c21">yt</font> holds <br/><font color="Maroon" title="c26">ft2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#T23" target="_self" title="TOPALG_5:th.23">Th23</a></span>;<br/>
<a NAME="E53:39"/>
<span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c21">yt</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">yt</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></span>;<br/>
<a NAME="E54:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A43</font></span>: 
<font color="Maroon" title="c26">ft2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c21">yt</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:39"><span class="lab"><font color="Green" title="E68">A42</font></span></a>, <a class="txt" href="#E45:39"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">sM1</font> =  <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font>;<br/>
<a NAME="E55:39"/><span class="lab"><font color="Green" title="E70">A44</font></span>: 
for <font color="Olive" title="b1">x</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>  holds  <font color="Maroon" title="c16">cP2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">x</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"> <font color="Maroon" title="c16">cP2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">x</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">x</font> = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c28">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c23">G</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c28">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="topalg_5.html#E5"><span class="lab"><font color="Green" title="E5">Lm5</font></span></a>, <a class="ref" href="binop_1.html#T30" title="BINOP_1:th.30">BINOP_1:30</a></span>
<br/>.= 
<font color="Maroon" title="c9">y0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E40:39"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c16">cP2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_3.html#T22" title="TOPALG_3:th.22">TOPALG_3:22</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="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c4">j0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c20">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c4">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:39"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="txt" href="#E31:39"><span class="lab"><font color="Green" title="E56">A30</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c21">yt</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="topalg_5.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a></span>
;<br/>
<a NAME="E57:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A45</font></span>: 
<font color="Maroon" title="c26">ft2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:39"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E52:39"><span class="lab"><font color="Green" title="E68">A42</font></span></a>, <a class="txt" href="#E55:39"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/>
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c23">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c4">j0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c20">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c4">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:39"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="txt" href="#E33:39"><span class="lab"><font color="Green" title="E57">A31</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c13">xt</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="topalg_5.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a></span>
;<br/>
<a NAME="E59:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A46</font></span>: 
<font color="Maroon" title="c19">ft1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c23">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:39"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E21:39"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="txt" href="#E43:39"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">Qt</font> =  <a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font>;<br/>
<span class="lab"><font color="Green" title="E73">A47</font></span>: <span class="p1">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c5">j1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D2" target="_self" title="TOPALG_5:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c23">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c13">xt</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:39"><span class="lab"><font color="Green" title="E72">A46</font></span></a>, <a class="txt" href="#E23:39"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c5">j1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D2" target="_self" title="TOPALG_5:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c21">yt</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:39"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="txt" href="#E54:39"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">Qt</font> =  <a href="topalg_5.html#K3" title="TOPALG_5:func.3">Prj1</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">xt</font>,<font color="Maroon" title="c21">yt</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:39"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E60:39"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E74">A48</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_12"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c30">s</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"> ( <font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c30">s</font>,<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="c22">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font> &amp; <font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c30">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">Qt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xt</font> &amp; <font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">yt</font> ) ) )</span><br/><a NAME="E1:39_12"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">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="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="topalg_5.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</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><font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c30">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c20">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c30">s</font>,<font color="Maroon" title="c4">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:39"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c22">ft</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="topalg_5.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a></span>
;<br/> <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="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c30">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">Qt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xt</font> &amp; <font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">yt</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E3:39_12"/>
<font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c30">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">Qt</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D2" target="_self" title="TOPALG_5:def.2">Def2</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">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xt</font> &amp; <font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">yt</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c31">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c31">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">xt</font> &amp; <font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c31">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">yt</font> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c31">t</font> = 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c4">j0</font>,<font color="Maroon" title="c23">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c13">xt</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E59:39"><span class="lab"><font color="Green" title="E72">A46</font></span></a>, <a class="txt" href="#E23:39"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</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="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c31">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">yt</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c23">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c31">t</font> = 
<span class="p1">(<span class="default"><a href="topalg_5.html#K4" title="TOPALG_5:func.4">Prj2</a> <font color="Maroon" title="c5">j1</font>,<font color="Maroon" title="c23">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_5.html#D3" target="_self" title="TOPALG_5:def.3">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c21">yt</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:39"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="txt" href="#E54:39"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="ref" href="topalg_3.html#T5" title="TOPALG_3:th.5">TOPALG_3:5</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="E64:39"/><span class="kw">then </span>
<font color="Maroon" title="c22">ft</font>,<font color="Maroon" title="c29">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:39"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">G</font> = <font color="Maroon" title="c23">G</font> as    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c22">ft</font>,<font color="Maroon" title="c29">Qt</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:39"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E63:39"><span class="lab"><font color="Green" title="E74">A48</font></span></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c22">ft</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">Qt</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">xt</font>,<font color="Maroon" title="c21">yt</font> ex <font color="Olive" title="b2">Ft</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c22">ft</font>,<font color="Olive" title="b1">Qt</font> st <br/>( <font color="Maroon" title="c22">ft</font>,<font color="Olive" title="b1">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">Ft</font> &amp; <font color="Maroon" title="c21">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b3">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c22">ft</font>,<font color="Olive" title="b1">Qt</font>  st <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">F1</font> holds <br/><font color="Olive" title="b2">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">F1</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c29">Qt</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">Ft</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c22">ft</font>,<font color="Maroon" title="c29">Qt</font> st <br/>( <font color="Maroon" title="c22">ft</font>,<font color="Maroon" title="c29">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">Ft</font> &amp; <font color="Maroon" title="c21">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b2">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c22">ft</font>,<font color="Maroon" title="c29">Qt</font>  st <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">F1</font> holds <br/><font color="Olive" title="b1">Ft</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">F1</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c30">G</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="c22">ft</font>,<font color="Maroon" title="c29">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a>  &amp; <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c30">G</font> &amp; <font color="Maroon" title="c21">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b1">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c22">ft</font>,<font color="Maroon" title="c29">Qt</font>  st <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">F1</font> holds <br/><font color="Maroon" title="c30">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">F1</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E66:39"/><span class="lab"><font color="Green" title="E75">A49</font></span>: 
<font color="Maroon" title="c22">ft</font>,<font color="Maroon" title="c29">Qt</font> <a href="borsuk_2.html#R4" title="BORSUK_2:pred.4">are_homotopic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:39"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E63:39"><span class="lab"><font color="Green" title="E74">A48</font></span></a>, <a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</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="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c30">G</font> &amp; <font color="Maroon" title="c21">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b1">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c22">ft</font>,<font color="Maroon" title="c29">Qt</font>  st <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">F1</font> holds <br/><font color="Maroon" title="c30">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">F1</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E67:39"/>
<font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c30">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:39"><span class="lab"><font color="Green" title="E61">A35</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="c21">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; ( for <font color="Olive" title="b1">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c22">ft</font>,<font color="Maroon" title="c29">Qt</font>  st <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">F1</font> holds <br/><font color="Maroon" title="c30">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">F1</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E68:39"/>
<font color="Maroon" title="c21">yt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">y0</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="txt" href="#E46:39"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</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">F1</font> being   <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c22">ft</font>,<font color="Maroon" title="c29">Qt</font>  st <font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">F1</font> holds <br/><font color="Maroon" title="c30">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">F1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c31">F1</font> be    <a href="borsuk_6.html#M1" title="BORSUK_6:mode.1">Homotopy</a> of <font color="Maroon" title="c22">ft</font>,<font color="Maroon" title="c29">Qt</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">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c31">F1</font> implies <font color="Maroon" title="c30">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c31">F1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E69:39"/><span class="lab"><font color="Green" title="E76">A50</font></span>: 
<font color="Maroon" title="c12">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c31">F1</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="c30">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c31">F1</font></span><br/>

<a NAME="E70:39"/><span class="lab"><font color="Green" title="E77">A51</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c31">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E71:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A52</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">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="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:39"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="txt" href="#E35:39"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<a NAME="E72:39"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">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="c32">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_13"/><span class="lab"><font color="Green" title="E79">A53</font></span>: 
<font color="Maroon" title="c32">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">x</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c33">x1</font>, <font color="Maroon" title="c34">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:39_13"/><span class="lab"><font color="Green" title="E80">A54</font></span>: 
<font color="Maroon" title="c33">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <span class="kw">and </span><br/><a NAME="E4:39_13"/><span class="lab"><font color="Green" title="E81">A55</font></span>: 
<font color="Maroon" title="c34">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:39_13"/><span class="lab"><font color="Green" title="E82">A56</font></span>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c33">x1</font>,<font color="Maroon" title="c34">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:39"><span class="lab"><font color="Green" title="E78">A52</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E6:39_13"/><span class="lab"><font color="Green" title="E83">A57</font></span>: 
<font color="Maroon" title="c34">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_13"><span class="lab"><font color="Green" title="E81">A55</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">x</font> = 
<font color="Maroon" title="c31">F1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c33">x1</font>,<font color="Maroon" title="c34">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_13"><span class="lab"><font color="Green" title="E79">A53</font></span></a>, <a class="txt" href="#E5:39_13"><span class="lab"><font color="Green" title="E82">A56</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<font color="Maroon" title="c22">ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c33">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:39"><span class="lab"><font color="Green" title="E75">A49</font></span></a>, <a class="txt" href="#E3:39_13"><span class="lab"><font color="Green" title="E80">A54</font></span></a>, <a class="txt" href="#E6:39_13"><span class="lab"><font color="Green" title="E83">A57</font></span></a>, <a class="ref" href="borsuk_6.html#D13" title="BORSUK_6:def.13">BORSUK_6:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c20">Ft</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c33">x1</font>,<font color="Maroon" title="c34">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E3:39_13"><span class="lab"><font color="Green" title="E80">A54</font></span></a>, <a class="txt" href="#E4:39_13"><span class="lab"><font color="Green" title="E81">A55</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c20">Ft</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c32">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_13"><span class="lab"><font color="Green" title="E82">A56</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="E73:39"/><span class="kw">then </span>
<font color="Maroon" title="c31">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Ft</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:39"><span class="lab"><font color="Green" title="E58">A32</font></span></a>, <a class="txt" href="#E35:39"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="txt" href="#E25:39"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E70:39"><span class="lab"><font color="Green" title="E77">A51</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/>
<span class="kw">hence </span><a NAME="E74:39"/>
<font color="Maroon" title="c30">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c31">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:39"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="txt" href="#E69:39"><span class="lab"><font color="Green" title="E76">A50</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>
