<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b3">P</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font><br/>  for <font color="Olive" title="b4">Q</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>  st <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/><span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Olive" title="b3">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b3">P</font>,<font color="Olive" title="b4">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font><br/>  for <font color="Olive" title="b2">Q</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>  st <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/><span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Olive" title="b1">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">S</font> = <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">P</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Q</font> being   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>  st <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/><span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font>,<font color="Olive" title="b1">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">Q</font> be   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  implies <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:104"/><span class="lab"><font color="Green" title="E84">A1</font></span>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">e2</font> =  <a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</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>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>) as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow12.html#T40" onmouseover="rs('yellow12/T40')" onmouseout="rh()">YELLOW12:40</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">gg</font> = <font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">e2</font>;<br/>
<a NAME="E3:104"/>
<font color="Maroon" title="c5">P</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">gg</font> = <font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">e2</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">S2</font> = <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c9">gg</font> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T9" onmouseover="rs('pre_topc/T9')" onmouseout="rh()">PRE_TOPC:9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c11">g</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T7" onmouseover="rs('topmetr/T7')" onmouseout="rh()">TOPMETR:7</a></span>;<br/>
<a NAME="E7:104"/><span class="lab"><font color="Green" title="E85">A2</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 <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span> holds  <font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span>)</span></span><br/>

<a NAME="E1:104_1"/>
<font color="Maroon" title="c13">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>
 
;<br/>
<a NAME="E2:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A3</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<a NAME="E3:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A4</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span>
 
;<br/>
<a NAME="E4:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A5</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="tarski.html#R2" title="TARSKI: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>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D2" onmouseover="rs('borsuk_1/D2')" onmouseout="rh()">BORSUK_1:def 2</a></span>;<br/>
<a NAME="E5:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A6</font></span>: 
<font color="Maroon" title="c13">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"><span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T21" onmouseover="rs('mcart_1/T21')" onmouseout="rh()">MCART_1:21</a></span>;<br/>
<a NAME="E6:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A7</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI: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">by</span> <span class="lab"><a class="txt" href="#E4:104_1"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E7:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A8</font></span>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="tarski.html#R2" title="TARSKI: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">by</span> <span class="lab"><a class="ref" href="jordan5b.html#T4" onmouseover="rs('jordan5b/T4')" onmouseout="rh()">JORDAN5B:4</a></span>;<br/>
<a NAME="E8:104_1"/>
<font color="Maroon" title="c13">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI: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">by</span> <span class="lab"><a class="txt" href="#E4:104_1"><span class="lab"><font color="Green" title="E88">A5</font></span></a>, <a class="txt" href="#E5:104_1"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E9:104_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A9</font></span>: 
<font color="Maroon" title="c7">e2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:104_1"><span class="lab"><font color="Green" title="E90">A7</font></span></a>, <a class="ref" href="funct_3.html#D5" onmouseover="rs('funct_3/D5')" onmouseout="rh()">FUNCT_3:def 5</a></span>;<br/>
<a NAME="E10:104_1"/><span class="lab"><font color="Green" title="E93">A10</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104_1"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> = 
<font color="Maroon" title="c9">gg</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104_1"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="txt" href="#E5:104_1"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:104_1"><span class="lab"><font color="Green" title="E89">A6</font></span></a>, <a class="txt" href="#E9:104_1"><span class="lab"><font color="Green" title="E92">A9</font></span></a>, <a class="txt" href="#E10:104_1"><span class="lab"><font color="Green" title="E93">A10</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E7:104_1"><span class="lab"><font color="Green" title="E91">A8</font></span></a>, <a class="ref" href="borsuk_2.html#D6" onmouseover="rs('borsuk_2/D6')" onmouseout="rh()">BORSUK_2:def 6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:104_1"/>
<font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c13">S3</font> = <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">S1</font> = <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">e1</font> =  <a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</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>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>) as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow12.html#T39" onmouseover="rs('yellow12/T39')" onmouseout="rh()">YELLOW12:39</a></span>;<br/>
<a NAME="E9:104"/><span class="lab"><font color="Green" title="E86">A11</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">PP</font> = <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">ff</font> = <font color="Maroon" title="c16">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">e1</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">e1</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T9" onmouseover="rs('pre_topc/T9')" onmouseout="rh()">PRE_TOPC:9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">f</font> = <font color="Maroon" title="c18">f</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T7" onmouseover="rs('topmetr/T7')" onmouseout="rh()">TOPMETR:7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">S12</font> = <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">S12</font> = <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> ;<br/>
<a NAME="E14:104"/><span class="lab"><font color="Green" title="E87">A12</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c21">S12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">hh</font> = <font color="Maroon" title="c16">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">e1</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">h</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">e1</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T9" onmouseover="rs('pre_topc/T9')" onmouseout="rh()">PRE_TOPC:9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">h</font> = <font color="Maroon" title="c23">h</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T7" onmouseover="rs('topmetr/T7')" onmouseout="rh()">TOPMETR:7</a></span>;<br/>
<a NAME="E17:104"/><span class="lab"><font color="Green" title="E88">A13</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 <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span> holds  <font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_2"><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 <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">h</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> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><br/>

<a NAME="E1:104_2"/>
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>
 
;<br/>
<a NAME="E2:104_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A14</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<a NAME="E3:104_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A15</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T60" target="_self" onmouseover="rs('borsuk_6/T60')" onmouseout="rh()">Th60</a></span>;<br/>
<a NAME="E4:104_2"/><span class="lab"><font color="Green" title="E91">A16</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104_2"><span class="lab"><font color="Green" title="E89">A14</font></span></a></span>;<br/>
<a NAME="E5:104_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A17</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI: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>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D2" onmouseover="rs('borsuk_1/D2')" onmouseout="rh()">BORSUK_1:def 2</a></span>;<br/>
<a NAME="E6:104_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A18</font></span>: 
<font color="Maroon" title="c25">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"><span class="p2">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T21" onmouseover="rs('mcart_1/T21')" onmouseout="rh()">MCART_1:21</a></span>;<br/>
<a NAME="E7:104_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A19</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI: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">by</span> <span class="lab"><a class="txt" href="#E5:104_2"><span class="lab"><font color="Green" title="E92">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E8:104_2"/>
<font color="Maroon" title="c25">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI: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">by</span> <span class="lab"><a class="txt" href="#E5:104_2"><span class="lab"><font color="Green" title="E92">A17</font></span></a>, <a class="txt" href="#E6:104_2"><span class="lab"><font color="Green" title="E93">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E9:104_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A20</font></span>: 
<font color="Maroon" title="c15">e1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104_2"><span class="lab"><font color="Green" title="E94">A19</font></span></a>, <a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>;<br/>
<a NAME="E10:104_2"/><span class="lab"><font color="Green" title="E96">A21</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">e1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:104_2"><span class="lab"><font color="Green" title="E91">A16</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<font color="Maroon" title="c24">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">e1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104_2"><span class="lab"><font color="Green" title="E89">A14</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">e1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:104_2"><span class="lab"><font color="Green" title="E93">A18</font></span></a>, <a class="txt" href="#E10:104_2"><span class="lab"><font color="Green" title="E96">A21</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E7:104_2"><span class="lab"><font color="Green" title="E94">A19</font></span></a>, <a class="txt" href="#E9:104_2"><span class="lab"><font color="Green" title="E95">A20</font></span></a>, <a class="txt" href="#E3:104_2"><span class="lab"><font color="Green" title="E90">A15</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:104_2"/>
<font color="Maroon" title="c24">h</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> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:104"/><span class="lab"><font color="Green" title="E89">A22</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 <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span> holds  <font color="Maroon" title="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_3"><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 <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">f</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="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span></span><br/>

<a NAME="E1:104_3"/>
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span>
 
;<br/>
<a NAME="E2:104_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A23</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<a NAME="E3:104_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A24</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T59" target="_self" onmouseover="rs('borsuk_6/T59')" onmouseout="rh()">Th59</a></span>;<br/>
<a NAME="E4:104_3"/><span class="lab"><font color="Green" title="E92">A25</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104_3"><span class="lab"><font color="Green" title="E90">A23</font></span></a></span>;<br/>
<a NAME="E5:104_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A26</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI: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>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D2" onmouseover="rs('borsuk_1/D2')" onmouseout="rh()">BORSUK_1:def 2</a></span>;<br/>
<a NAME="E6:104_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A27</font></span>: 
<font color="Maroon" title="c25">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"><span class="p2">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T21" onmouseover="rs('mcart_1/T21')" onmouseout="rh()">MCART_1:21</a></span>;<br/>
<a NAME="E7:104_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A28</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI: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">by</span> <span class="lab"><a class="txt" href="#E5:104_3"><span class="lab"><font color="Green" title="E93">A26</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E8:104_3"/>
<font color="Maroon" title="c25">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI: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">by</span> <span class="lab"><a class="txt" href="#E5:104_3"><span class="lab"><font color="Green" title="E93">A26</font></span></a>, <a class="txt" href="#E6:104_3"><span class="lab"><font color="Green" title="E94">A27</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E9:104_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A29</font></span>: 
<font color="Maroon" title="c15">e1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104_3"><span class="lab"><font color="Green" title="E95">A28</font></span></a>, <a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>;<br/>
<a NAME="E10:104_3"/><span class="lab"><font color="Green" title="E97">A30</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">e1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:104_3"><span class="lab"><font color="Green" title="E92">A25</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<font color="Maroon" title="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">PP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">e1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104_3"><span class="lab"><font color="Green" title="E90">A23</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">e1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:104_3"><span class="lab"><font color="Green" title="E97">A30</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E6:104_3"><span class="lab"><font color="Green" title="E94">A27</font></span></a>, <a class="txt" href="#E7:104_3"><span class="lab"><font color="Green" title="E95">A28</font></span></a>, <a class="txt" href="#E9:104_3"><span class="lab"><font color="Green" title="E96">A29</font></span></a>, <a class="txt" href="#E3:104_3"><span class="lab"><font color="Green" title="E91">A24</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:104_3"/>
<font color="Maroon" title="c19">f</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="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:104"/><span class="lab"><font color="Green" title="E90">A31</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span></span>)</span> implies <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:104_4"/>
<font color="Maroon" title="c25">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">p</font></span><br/>

<a NAME="E2:104_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A32</font></span>: 
<font color="Maroon" title="c25">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E3:104_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A33</font></span>: 
<font color="Maroon" title="c25">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">r</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span><span class="kw"> such that </span><br/><a NAME="E5:104_4"/><span class="lab"><font color="Green" title="E93">A34</font></span>: 
<font color="Maroon" title="c26">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font>
 <span class="kw">and </span><br/><a NAME="E6:104_4"/><span class="lab"><font color="Green" title="E94">A35</font></span>: 
<font color="Maroon" title="c26">r</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">r</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T57" target="_self" onmouseover="rs('borsuk_6/T57')" onmouseout="rh()">Th57</a></span>;<br/>
<a NAME="E7:104_4"/><span class="lab"><font color="Green" title="E95">A36</font></span>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:104_4"><span class="lab"><font color="Green" title="E94">A35</font></span></a></span>;<br/>
<a NAME="E8:104_4"/>
<font color="Maroon" title="c25">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104_4"><span class="lab"><font color="Green" title="E92">A33</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">pp</font> = <font color="Maroon" title="c25">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<a NAME="E10:104_4"/>
<font color="Maroon" title="c25">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104_4"><span class="lab"><font color="Green" title="E91">A32</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E11:104_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A37</font></span>: 
<font color="Maroon" title="c27">pp</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">p</font> = 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">pp</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:104"><span class="lab"><font color="Green" title="E89">A22</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E5:104_4"><span class="lab"><font color="Green" title="E93">A34</font></span></a>, <a class="txt" href="#E7:104_4"><span class="lab"><font color="Green" title="E95">A36</font></span></a>, <a class="txt" href="#E11:104_4"><span class="lab"><font color="Green" title="E96">A37</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:104_4"/>
<font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c25">s3</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E21:104"/><span class="lab"><font color="Green" title="E91">A38</font></span>: 
<font color="Maroon" title="c25">s3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E22:104"/><span class="lab"><font color="Green" title="E92">A39</font></span>: 
( <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c21">S12</font> &amp; <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c21">S12</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T22" onmouseover="rs('topmetr/T22')" onmouseout="rh()">TOPMETR:22</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E23:104"/><span class="lab"><font color="Green" title="E93">A40</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E24:104"/><span class="lab"><font color="Green" title="E94">A41</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">s1</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span>, <font color="Maroon" title="c27">s2</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c21">S12</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:104"><span class="lab"><font color="Green" title="E87">A12</font></span></a>, <a class="txt" href="#E23:104"><span class="lab"><font color="Green" title="E93">A40</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E26:104"/><span class="lab"><font color="Green" title="E95">A42</font></span>: 
<font color="Maroon" title="c26">s1</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:104"><span class="lab"><font color="Green" title="E94">A41</font></span></a>, <a class="ref" href="tops_2.html#T26" onmouseover="rs('tops_2/T26')" onmouseout="rh()">TOPS_2:26</a></span>;<br/>
<a NAME="E27:104"/><span class="lab"><font color="Green" title="E96">A43</font></span>: 
<font color="Maroon" title="c27">s2</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:104"><span class="lab"><font color="Green" title="E93">A40</font></span></a>, <a class="ref" href="tops_2.html#T26" onmouseover="rs('tops_2/T26')" onmouseout="rh()">TOPS_2:26</a></span>;<br/>
<a NAME="E28:104"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c21">S12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:104"><span class="lab"><font color="Green" title="E87">A12</font></span></a>, <a class="txt" href="#E23:104"><span class="lab"><font color="Green" title="E93">A40</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">fg</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c21">S12</font>,<font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E30:104"/><span class="lab"><font color="Green" title="E97">A44</font></span>: 
<font color="Maroon" title="c28">fg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c12">g</font>
 <span class="kw">and </span><br/><a NAME="E31:104"/><span class="lab"><font color="Green" title="E98">A45</font></span>: 
<font color="Maroon" title="c28">fg</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:104"><span class="lab"><font color="Green" title="E90">A31</font></span></a>, <a class="txt" href="#E22:104"><span class="lab"><font color="Green" title="E92">A39</font></span></a>, <a class="txt" href="#E26:104"><span class="lab"><font color="Green" title="E95">A42</font></span></a>, <a class="txt" href="#E27:104"><span class="lab"><font color="Green" title="E96">A43</font></span></a>, <a class="ref" href="jgraph_2.html#T1" onmouseover="rs('jgraph_2/T1')" onmouseout="rh()">JGRAPH_2:1</a></span>;<br/>
<a NAME="E32:104"/><span class="lab"><font color="Green" title="E99">A46</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E33:104"/><span class="lab"><font color="Green" title="E100">A47</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c21">S12</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c28">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c29">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c21">S12</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span></span>)</span> implies <font color="Maroon" title="c28">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">p</font> )</span><br/>

<a NAME="E1:104_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T66" target="_self" onmouseover="rs('borsuk_6/T66')" onmouseout="rh()">Th66</a>, <a class="ref" href="borsuk_6.html#T67" target="_self" onmouseover="rs('borsuk_6/T67')" onmouseout="rh()">Th67</a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E2:104_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A48</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:104_5"/>
<font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c21">S12</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c28">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">p</font></span><br/>

<a NAME="E4:104_5"/><span class="kw">then </span>
<font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:104"><span class="lab"><font color="Green" title="E87">A12</font></span></a>, <a class="txt" href="#E32:104"><span class="lab"><font color="Green" title="E99">A46</font></span></a>, <a class="ref" href="borsuk_6.html#T72" target="_self" onmouseover="rs('borsuk_6/T72')" onmouseout="rh()">Th72</a>, <a class="ref" href="xboole_1.html#T23" onmouseover="rs('xboole_1/T23')" onmouseout="rh()">XBOOLE_1:23</a></span>;<br/>
<a NAME="E5:104_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A49</font></span>: 
<font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:104_5"><span class="lab"><font color="Green" title="E101">A48</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>;<br/>
<a NAME="E6:104_5"/><span class="kw">then </span>
<font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">pp</font> = <font color="Maroon" title="c29">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<a NAME="E8:104_5"/><span class="lab"><font color="Green" title="E103">A50</font></span>: 
 ex <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p</font> &amp; <font color="Olive" title="b1">q</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:104_5"><span class="lab"><font color="Green" title="E102">A49</font></span></a>, <a class="ref" href="borsuk_6.html#T58" target="_self" onmouseover="rs('borsuk_6/T58')" onmouseout="rh()">Th58</a></span>;<br/>
<a NAME="E9:104_5"/>
<font color="Maroon" title="c30">pp</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:104_5"><span class="lab"><font color="Green" title="E102">A49</font></span></a>, <a class="ref" href="borsuk_6.html#T27" target="_self" onmouseover="rs('borsuk_6/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<a NAME="E10:104_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A51</font></span>: 
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">pp</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> <a href="tarski.html#R2" title="TARSKI: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">by</span> <span class="lab"><a class="ref" href="jordan5b.html#T4" onmouseover="rs('jordan5b/T4')" onmouseout="rh()">JORDAN5B:4</a></span>;<br/>
<a NAME="E11:104_5"/><span class="lab"><font color="Green" title="E105">A52</font></span>: 
<font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:104_5"><span class="lab"><font color="Green" title="E102">A49</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E12:104_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A53</font></span>: 
<font color="Maroon" title="c30">pp</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<a NAME="E13:104_5"/>
<font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:104_5"><span class="lab"><font color="Green" title="E105">A52</font></span></a>, <a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<a NAME="E14:104_5"/><span class="kw">then </span>
<font color="Maroon" title="c29">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c28">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">p</font> = 
<font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:104"><span class="lab"><font color="Green" title="E97">A44</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">pp</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E12:104_5"><span class="lab"><font color="Green" title="E106">A53</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c30">pp</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E10:104_5"><span class="lab"><font color="Green" title="E104">A51</font></span></a>, <a class="ref" href="borsuk_2.html#D6" onmouseover="rs('borsuk_2/D6')" onmouseout="rh()">BORSUK_2:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:104"><span class="lab"><font color="Green" title="E88">A13</font></span></a>, <a class="txt" href="#E8:104_5"><span class="lab"><font color="Green" title="E103">A50</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:104_5"/>
<font color="Maroon" title="c28">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E34:104"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c21">S12</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:104"><span class="lab"><font color="Green" title="E87">A12</font></span></a>, <a class="txt" href="#E32:104"><span class="lab"><font color="Green" title="E99">A46</font></span></a>, <a class="ref" href="borsuk_6.html#T56" target="_self" onmouseover="rs('borsuk_6/T56')" onmouseout="rh()">Th56</a>, <a class="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="borsuk_1.html#D2" onmouseover="rs('borsuk_1/D2')" onmouseout="rh()">BORSUK_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">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#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span>,<font color="Maroon" title="c1">T</font><span class="kw"> such that </span><br/><a NAME="E36:104"/><span class="lab"><font color="Green" title="E101">A54</font></span>: 
<font color="Maroon" title="c29">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">fg</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c24">h</font>
 <span class="kw">and </span><br/><a NAME="E37:104"/><span class="lab"><font color="Green" title="E102">A55</font></span>: 
<font color="Maroon" title="c29">H</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:104"><span class="lab"><font color="Green" title="E87">A12</font></span></a>, <a class="txt" href="#E31:104"><span class="lab"><font color="Green" title="E98">A45</font></span></a>, <a class="txt" href="#E33:104"><span class="lab"><font color="Green" title="E100">A47</font></span></a>, <a class="txt" href="#E21:104"><span class="lab"><font color="Green" title="E91">A38</font></span></a>, <a class="ref" href="jgraph_2.html#T1" onmouseover="rs('jgraph_2/T1')" onmouseout="rh()">JGRAPH_2:1</a></span>;<br/>
<a NAME="E38:104"/><span class="lab"><font color="Green" title="E103">A56</font></span>: 
 for <font color="Olive" title="b1">s</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="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">s</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_6"><span class="kw">proof </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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font> &amp; <font color="Maroon" title="c29">H</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="c6">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:104_6"/>
<font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">H</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="c6">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_6_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:104_6_1"/><span class="lab"><font color="Green" title="E104">A57</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">s</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_6_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:104_6_1_1"/>
( <font color="Maroon" title="c30">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 or <font color="Maroon" title="c30">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 or <font color="Maroon" title="c30">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_6_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:104_6_1_1_1"/><span class="lab"><font color="Green" title="E105">A58</font></span>: 
<font color="Maroon" title="c30">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font></span><br/><div class="add"><a NAME="E2:104_6_1_1_1"/><span class="kw">then </span>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T71" target="_self" onmouseover="rs('borsuk_6/T71')" onmouseout="rh()">Th71</a></span>;<br/><a NAME="E3:104_6_1_1_1"/><span class="kw">then </span>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/><a NAME="E4:104_6_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A59</font></span>: 
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">g</font>
 ;<br/><a NAME="E5:104_6_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104_6_1_1_1"><span class="lab"><font color="Green" title="E105">A58</font></span></a>, <a class="ref" href="borsuk_6.html#T70" target="_self" onmouseover="rs('borsuk_6/T70')" onmouseout="rh()">Th70</a></span>;<br/><a NAME="E6:104_6_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A60</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/><a NAME="E7:104_6_1_1_1"/>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104_6_1_1_1"><span class="lab"><font color="Green" title="E105">A58</font></span></a>, <a class="ref" href="borsuk_6.html#T71" target="_self" onmouseover="rs('borsuk_6/T71')" onmouseout="rh()">Th71</a></span>;<br/><a NAME="E8:104_6_1_1_1"/><span class="kw">then </span>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/><a NAME="E9:104_6_1_1_1"/><span class="kw">then </span>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c24">h</font>
 ;<br/><span class="kw">then </span><font color="Maroon" title="c29">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c28">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:104"><span class="lab"><font color="Green" title="E101">A54</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>
<br/>.= 
<font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:104"><span class="lab"><font color="Green" title="E97">A44</font></span></a>, <a class="txt" href="#E4:104_6_1_1_1"><span class="lab"><font color="Green" title="E106">A59</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:104"><span class="lab"><font color="Green" title="E89">A22</font></span></a>, <a class="txt" href="#E1:104_6_1"><span class="lab"><font color="Green" title="E104">A57</font></span></a>, <a class="txt" href="#E6:104_6_1_1_1"><span class="lab"><font color="Green" title="E107">A60</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <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="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E1:104_6_1_1_1"><span class="lab"><font color="Green" title="E105">A58</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E11:104_6_1_1_1"/>
<font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_6_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:104_6_1_1_2"/><span class="lab"><font color="Green" title="E105">A61</font></span>: 
<font color="Maroon" title="c30">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font></span><br/><div class="add"><a NAME="E2:104_6_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A62</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T66" target="_self" onmouseover="rs('borsuk_6/T66')" onmouseout="rh()">Th66</a>, <a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/><a NAME="E3:104_6_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c24">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c29">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:104"><span class="lab"><font color="Green" title="E101">A54</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:104"><span class="lab"><font color="Green" title="E88">A13</font></span></a>, <a class="txt" href="#E1:104_6_1"><span class="lab"><font color="Green" title="E104">A57</font></span></a>, <a class="txt" href="#E2:104_6_1_1_2"><span class="lab"><font color="Green" title="E106">A62</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E1:104_6_1_1_2"><span class="lab"><font color="Green" title="E105">A61</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <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="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E1:104_6_1_1_2"><span class="lab"><font color="Green" title="E105">A61</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:104_6_1_1_2"/>
<font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_6_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:104_6_1_1_3"/><span class="lab"><font color="Green" title="E105">A63</font></span>: 
<font color="Maroon" title="c30">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font></span><br/><div class="add"><a NAME="E2:104_6_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T69" target="_self" onmouseover="rs('borsuk_6/T69')" onmouseout="rh()">Th69</a></span>;<br/><a NAME="E3:104_6_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A64</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/><a NAME="E4:104_6_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c24">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c29">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:104"><span class="lab"><font color="Green" title="E101">A54</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:104"><span class="lab"><font color="Green" title="E88">A13</font></span></a>, <a class="txt" href="#E1:104_6_1"><span class="lab"><font color="Green" title="E104">A57</font></span></a>, <a class="txt" href="#E3:104_6_1_1_3"><span class="lab"><font color="Green" title="E106">A64</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <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="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E1:104_6_1_1_3"><span class="lab"><font color="Green" title="E105">A63</font></span></a>, <a class="ref" href="borsuk_2.html#D5" onmouseover="rs('borsuk_2/D5')" onmouseout="rh()">BORSUK_2:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:104_6_1_1_3"/>
<font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c30">s</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:104_6"/>
<font color="Maroon" title="c29">H</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="c6">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_6_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:104_6_2"/><span class="lab"><font color="Green" title="E104">A65</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<a NAME="E2:104_6_2"/><span class="lab"><font color="Green" title="E105">A66</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">s</font>
 
;<br/>
<a NAME="E3:104_6_2"/><span class="lab"><font color="Green" title="E106">A67</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:104_6_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A68</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_6_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:104_6_2_1"/>
( <font color="Maroon" title="c30">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or <font color="Maroon" title="c30">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_6_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:104_6_2_1_1"/><span class="lab"><font color="Green" title="E108">A69</font></span>: 
<font color="Maroon" title="c30">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">H</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="c6">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font></span><br/><div class="add"><a NAME="E2:104_6_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T65" target="_self" onmouseover="rs('borsuk_6/T65')" onmouseout="rh()">Th65</a></span>;<br/><a NAME="E3:104_6_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A70</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/><a NAME="E4:104_6_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A71</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E5:104_6_2_1_1"/>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104_6_2_1_1"><span class="lab"><font color="Green" title="E108">A69</font></span></a>, <a class="ref" href="borsuk_6.html#T63" target="_self" onmouseover="rs('borsuk_6/T63')" onmouseout="rh()">Th63</a></span>;<br/><a NAME="E6:104_6_2_1_1"/><span class="kw">then </span>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/><a NAME="E7:104_6_2_1_1"/><span class="kw">then </span>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c24">h</font>
 ;<br/><span class="kw">then </span><font color="Maroon" title="c29">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c28">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:104"><span class="lab"><font color="Green" title="E101">A54</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>
<br/>.= 
<font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:104"><span class="lab"><font color="Green" title="E97">A44</font></span></a>, <a class="txt" href="#E4:104_6_2_1_1"><span class="lab"><font color="Green" title="E110">A71</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E1:104_6_2"><span class="lab"><font color="Green" title="E104">A65</font></span></a>, <a class="txt" href="#E3:104_6_2_1_1"><span class="lab"><font color="Green" title="E109">A70</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:104"><span class="lab"><font color="Green" title="E86">A11</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c6">Q</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="#E3:104_6_2"><span class="lab"><font color="Green" title="E106">A67</font></span></a>, <a class="txt" href="#E4:104_6_2"><span class="lab"><font color="Green" title="E107">A68</font></span></a>, <a class="ref" href="funct_1.html#D10" onmouseover="rs('funct_1/D10')" onmouseout="rh()">FUNCT_1:def 10</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:104_6_2_1_1"/>
<font color="Maroon" title="c29">H</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="c6">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_6_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:104_6_2_1_2"/><span class="lab"><font color="Green" title="E108">A72</font></span>: 
<font color="Maroon" title="c30">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">H</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="c6">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font></span><br/><div class="add"><a NAME="E2:104_6_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A73</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T66" target="_self" onmouseover="rs('borsuk_6/T66')" onmouseout="rh()">Th66</a>, <a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/><a NAME="E3:104_6_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c24">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c29">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:104"><span class="lab"><font color="Green" title="E101">A54</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:104"><span class="lab"><font color="Green" title="E88">A13</font></span></a>, <a class="txt" href="#E2:104_6_2"><span class="lab"><font color="Green" title="E105">A66</font></span></a>, <a class="txt" href="#E2:104_6_2_1_2"><span class="lab"><font color="Green" title="E109">A73</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E1:104_6_2_1_2"><span class="lab"><font color="Green" title="E108">A72</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:104"><span class="lab"><font color="Green" title="E86">A11</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c6">Q</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="#E3:104_6_2"><span class="lab"><font color="Green" title="E106">A67</font></span></a>, <a class="txt" href="#E4:104_6_2"><span class="lab"><font color="Green" title="E107">A68</font></span></a>, <a class="ref" href="funct_1.html#D10" onmouseover="rs('funct_1/D10')" onmouseout="rh()">FUNCT_1:def 10</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:104_6_2_1_2"/>
<font color="Maroon" title="c29">H</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="c6">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E39:104"/>
 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="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c29">H</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="c2">a</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c30">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c30">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c30">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:104_7"/>
<font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c30">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c30">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_7_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:104_7_1"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI: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">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c30">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="borsuk_6.html#E83"><span class="lab"><font color="Green" title="E76">Lm1</font></span></a></span>;<br/>
<a NAME="E3:104_7_1"/>
<font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T61" target="_self" onmouseover="rs('borsuk_6/T61')" onmouseout="rh()">Th61</a></span>;<br/>
<a NAME="E4:104_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A74</font></span>: 
<font color="Maroon" title="c31">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK9" title="BORSUK_6:NK.9">IAA</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<a NAME="E5:104_7_1"/>
<font color="Maroon" title="c31">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E6:104_7_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T60" target="_self" onmouseover="rs('borsuk_6/T60')" onmouseout="rh()">Th60</a></span>;<br/>
<a NAME="E7:104_7_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<a NAME="E8:104_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A75</font></span>: 
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c30">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c24">h</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_7_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:104_7_1_1"/>
( <font color="Maroon" title="c30">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 or <font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_7_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:104_7_1_1_1"/>
<font color="Maroon" title="c30">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c30">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/><div class="add"><a NAME="E2:104_7_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T62" target="_self" onmouseover="rs('borsuk_6/T62')" onmouseout="rh()">Th62</a></span>;<br/><a NAME="E3:104_7_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/><a NAME="E4:104_7_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">g</font>
 ;<br/><span class="kw">then </span><font color="Maroon" title="c28">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c30">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c19">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c30">t</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="#E30:104"><span class="lab"><font color="Green" title="E97">A44</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:104"><span class="lab"><font color="Green" title="E89">A22</font></span></a>, <a class="txt" href="#E4:104_7_1"><span class="lab"><font color="Green" title="E104">A74</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:104_7_1_1_1"/>
<font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c30">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:104"><span class="lab"><font color="Green" title="E101">A54</font></span></a>, <a class="txt" href="#E8:104_7_1"><span class="lab"><font color="Green" title="E105">A75</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_7_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:104_7_1_1_2"/><span class="lab"><font color="Green" title="E106">A76</font></span>: 
<font color="Maroon" title="c30">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c30">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/><div class="add"><a NAME="E2:104_7_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A77</font></span>: 
<font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK11" title="BORSUK_6:NK.11">IBB</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T64" target="_self" onmouseover="rs('borsuk_6/T64')" onmouseout="rh()">Th64</a>, <a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/><a NAME="E3:104_7_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c28">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c30">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:104"><span class="lab"><font color="Green" title="E97">A44</font></span></a>, <a class="txt" href="#E1:104_7_1_1_2"><span class="lab"><font color="Green" title="E106">A76</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:104"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E1:104_7_1_1_2"><span class="lab"><font color="Green" title="E106">A76</font></span></a>, <a class="txt" href="#E2:104_7_1_1_2"><span class="lab"><font color="Green" title="E107">A77</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E1:104_7_1_1_2"><span class="lab"><font color="Green" title="E106">A76</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:104_7_1_1_2"/>
<font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c30">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:104"><span class="lab"><font color="Green" title="E101">A54</font></span></a>, <a class="txt" href="#E8:104_7_1"><span class="lab"><font color="Green" title="E105">A75</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:104_7"/>
<font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c30">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="104_7_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:104_7_2"/>
1 <a href="tarski.html#R2" title="TARSKI: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">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T43" onmouseover="rs('borsuk_1/T43')" onmouseout="rh()">BORSUK_1:43</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c30">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="borsuk_6.html#E83"><span class="lab"><font color="Green" title="E76">Lm1</font></span></a></span>;<br/>
<a NAME="E3:104_7_2"/>
<font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_6.html#T68" target="_self" onmouseover="rs('borsuk_6/T68')" onmouseout="rh()">Th68</a></span>;<br/>
<a NAME="E4:104_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A78</font></span>: 
<font color="Maroon" title="c31">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</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#K2" title="BORSUK_1:func.2">:]</a></span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="borsuk_6.html#NK13" title="BORSUK_6:NK.13">ICC</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" onmouseover="rs('pre_topc/T8')" onmouseout="rh()">PRE_TOPC:8</a></span>;<br/>
<a NAME="E5:104_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A79</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c30">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c24">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<font color="Maroon" title="c24">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c30">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c31">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:104"><span class="lab"><font color="Green" title="E88">A13</font></span></a>, <a class="txt" href="#E4:104_7_2"><span class="lab"><font color="Green" title="E104">A78</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:104_7_2"/>
<font color="Maroon" title="c29">H</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<font color="Maroon" title="c30">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:104"><span class="lab"><font color="Green" title="E101">A54</font></span></a>, <a class="txt" href="#E5:104_7_2"><span class="lab"><font color="Green" title="E105">A79</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E40:104"/>
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:104"><span class="lab"><font color="Green" title="E102">A55</font></span></a>, <a class="txt" href="#E38:104"><span class="lab"><font color="Green" title="E103">A56</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
