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<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#L1" title="PRE_TOPC:struct.1">TopStruct</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">c1</font>, <font color="Olive" title="b2">c2</font> being   <a href="topalg_6.html#V7" title="TOPALG_6:attr.7">with_endpoints</a>  <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">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="b5">p1</font>, <font color="Olive" title="b6">p2</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>  st <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">p1</font> &amp; <font color="Olive" title="b2">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">p2</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a>  holds <br/>( <font color="Olive" title="b1">c1</font>,<font color="Olive" title="b2">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a>  iff <font color="Olive" title="b5">p1</font>,<font color="Olive" title="b6">p2</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">c1</font>, <font color="Maroon" title="c3">c2</font> be   <a href="topalg_6.html#V7" title="TOPALG_6:attr.7">with_endpoints</a>  <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</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">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">p1</font>, <font color="Olive" title="b4">p2</font> being    <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="b2">b</font>  st <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p1</font> &amp; <font color="Maroon" title="c3">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">p2</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a>  holds <br/>( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a>  iff <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</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="c4">a</font>, <font color="Maroon" title="c5">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">p1</font>, <font color="Olive" title="b2">p2</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>  st <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Maroon" title="c3">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a>  holds <br/>( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a>  iff <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</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">p1</font>, <font color="Maroon" title="c7">p2</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</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="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> &amp; <font color="Maroon" title="c3">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a>  implies ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a>  iff <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:65"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
( <font color="Maroon" title="c2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> &amp; <font color="Maroon" title="c3">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</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"> (  not <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a>  or ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a>  iff <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:65"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="borsuk_2.html#R1" title="BORSUK_2: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"> ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a>  iff <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )</span><br/>

<a NAME="E3:65"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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> &amp; 1 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="ref" href="borsuk_1.html#T40" onmouseover="rs('borsuk_1/T40')" onmouseout="rh()">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E4:65"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
(  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="topalg_6.html#T4" target="_self" onmouseover="rs('topalg_6/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E5:65"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">p1</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> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">p2</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/>
<span class="kw">thus </span><a NAME="E6:65"/>
( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a>  implies <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</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="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  implies <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:65_1"/>
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</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="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">aa</font>, <font color="Maroon" title="c9">bb</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c10">pp1</font>, <font color="Maroon" title="c11">pp2</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c8">aa</font>,<font color="Maroon" title="c9">bb</font><span class="kw"> such that </span><br/><a NAME="E3:65_1"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
( <font color="Maroon" title="c10">pp1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">c1</font></span>)</span></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c11">pp2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<span class="p2">(<span class="default"><a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">sup</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">c2</font></span>)</span></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c10">pp1</font>,<font color="Maroon" title="c11">pp2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a>  )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#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="E5:65_1"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
( <font color="Maroon" title="c12">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; (  for <font color="Olive" title="b1">t</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> holds <br/> ( <font color="Maroon" title="c12">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">pp1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c12">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">t</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pp2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c12">f</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="c8">aa</font> &amp; <font color="Maroon" title="c12">f</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="c9">bb</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a></span>;<br/>
<a NAME="E6:65_1"/><span class="lab"><font color="Green" title="E44">A8</font></span>: 
( <font color="Maroon" title="c10">pp1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">p1</font> &amp; <font color="Maroon" title="c11">pp2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">p2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:65_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="topalg_6.html#T1" target="_self" onmouseover="rs('topalg_6/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="relat_1.html#T52" onmouseover="rs('relat_1/T52')" onmouseout="rh()">RELAT_1:52</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/>
<a NAME="E7:65_1"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
( <font color="Maroon" title="c12">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">pp1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> &amp; <font color="Maroon" title="c12">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pp2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> &amp; <font color="Maroon" title="c12">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">aa</font> &amp; <font color="Maroon" title="c12">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">aa</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E39">A3</font></span></a></span>;<br/>
<a NAME="E8:65_1"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
( <font color="Maroon" title="c12">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">pp1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c12">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pp2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c12">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">bb</font> &amp; <font color="Maroon" title="c12">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (1,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">bb</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E39">A3</font></span></a></span>;<br/>
<a NAME="E9:65_1"/>
( <font color="Maroon" title="c8">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c9">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65_1"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="txt" href="#E7:65_1"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="txt" href="#E8:65_1"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E38">A2</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="E10:65_1"/>
<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</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="#E5:65_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E6:65_1"><span class="lab"><font color="Green" title="E44">A8</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><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E7:65"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</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="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a> </span><br/>

<a NAME="E8:65"/>
( <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> &amp; <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K1" title="BORSUK_6:func.1">L[01]</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="topalg_6.html#T1" target="_self" onmouseover="rs('topalg_6/T1')" onmouseout="rh()">Th1</a>, <a class="ref" href="relat_1.html#T52" onmouseover="rs('relat_1/T52')" onmouseout="rh()">RELAT_1:52</a>, <a class="ref" href="topmetr.html#T20" onmouseover="rs('topmetr/T20')" onmouseout="rh()">TOPMETR:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:65"/>
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="topalg_6.html#R1" title="TOPALG_6:pred.1">are_homotopic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E7:65"><span class="lab"><font color="Green" title="E42">A11</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>
