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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#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"> ( <font color="Maroon" title="c1">T</font> is  <a href="topalg_6.html#V2" title="TOPALG_6:attr.2">simply_connected</a>  iff  for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; (  for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> holds   <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font>)</span>)</span>,<font color="Olive" title="b3">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font>)</span>)</span>,<font color="Olive" title="b4">Q</font>) ) ) )</span><br/>

<div><span class="kw">hereby </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"> ( (  for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; (  for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> holds   <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font>)</span>)</span>,<font color="Olive" title="b3">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font>)</span>)</span>,<font color="Olive" title="b4">Q</font>) ) ) ) implies <font color="Maroon" title="c1">T</font> is  <a href="topalg_6.html#V2" title="TOPALG_6:attr.2">simply_connected</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c1">T</font> is  <a href="topalg_6.html#V2" title="TOPALG_6:attr.2">simply_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">  for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; (  for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> holds   <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font>)</span>)</span>,<font color="Olive" title="b3">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font>)</span>)</span>,<font color="Olive" title="b4">Q</font>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">t1</font>, <font color="Maroon" title="c3">t2</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"> ( <font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; (  for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font> holds   <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font>)</span>)</span>,<font color="Olive" title="b1">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font>)</span>)</span>,<font color="Olive" title="b2">Q</font>) ) )</span><br/><span class="kw">thus </span><a NAME="E2:20_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="borsuk_2.html#D3" onmouseover="rs('borsuk_2/D3')" onmouseout="rh()">BORSUK_2:def 3</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">  for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font> holds   <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font>)</span>)</span>,<font color="Olive" title="b1">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font>)</span>)</span>,<font color="Olive" title="b2">Q</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</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">  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font>)</span>)</span>,<font color="Maroon" title="c4">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font>)</span>)</span>,<font color="Maroon" title="c5">Q</font>)</span><br/><span class="kw">set </span><font color="Maroon" title="c6">E</font> =  <a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font>);<br/><a NAME="E3:20_1"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c4">P</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">Q</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">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="#E1:20_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="topalg_1.html#T22" onmouseover="rs('topalg_1/T22')" onmouseout="rh()">TOPALG_1:22</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">C</font> =  the   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">t1</font>;<br/><a NAME="E4:20_1"/>
<font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.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">Q</font></span>)</span>, the   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">t1</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="#E1:20_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="topalg_6.html#T11" target="_self" onmouseover="rs('topalg_6/T11')" onmouseout="rh()">Th11</a></span>;<br/><a NAME="E5:20_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">Q</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">Q</font>, the   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">t1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">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="#E1:20_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="borsuk_6.html#T76" onmouseover="rs('borsuk_6/T76')" onmouseout="rh()">BORSUK_6:76</a></span>;<br/><a NAME="E6:20_1"/>
 the   <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">t1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c5">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="#E1:20_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="borsuk_6.html#T83" onmouseover="rs('borsuk_6/T83')" onmouseout="rh()">BORSUK_6:83</a></span>;<br/><a NAME="E7:20_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c5">Q</font></span>)</span></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c5">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="#E5:20_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="borsuk_6.html#T79" onmouseover="rs('borsuk_6/T79')" onmouseout="rh()">BORSUK_6:79</a></span>;<br/><a NAME="E8:20_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">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="#E3:20_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="borsuk_6.html#T79" onmouseover="rs('borsuk_6/T79')" onmouseout="rh()">BORSUK_6:79</a></span>;<br/><span class="kw">hence </span><a NAME="E9:20_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font>)</span>)</span>,<font color="Maroon" title="c4">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font>)</span>)</span>,<font color="Maroon" title="c5">Q</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="topalg_1.html#T46" onmouseover="rs('topalg_1/T46')" onmouseout="rh()">TOPALG_1:46</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="E2:20"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  &amp; (  for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> holds   <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font>)</span>)</span>,<font color="Olive" title="b3">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K3" title="TOPALG_1:func.3">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font>)</span>)</span>,<font color="Olive" title="b4">Q</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="c1">T</font> is  <a href="topalg_6.html#V2" title="TOPALG_6:attr.2">simply_connected</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:20"/>
<font color="Maroon" title="c1">T</font> is  <a href="topalg_6.html#V1" title="TOPALG_6:attr.1">having_trivial_Fundamental_Group</a> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topalg_6.html#D2" target="_self" onmouseover="rs('topalg_6/D2')" onmouseout="rh()">TOPALG_6:def 2</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="c1">T</font> is  <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">pathwise_connected</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topalg_6.html#D1" target="_self" onmouseover="rs('topalg_6/D1')" onmouseout="rh()">TOPALG_6:def 1</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">  <a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>) is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="topalg_1.html#NK6" title="TOPALG_1:NK.6">pi_1</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>)</span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="struct_0.html#D10" onmouseover="rs('struct_0/D10')" onmouseout="rh()">STRUCT_0:def 10</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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font></span><br/>



<a NAME="E1:20_2"/>
(  ex <font color="Olive" title="b1">P</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">t</font> st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>)</span>)</span>,<font color="Olive" title="b1">P</font>) &amp;  ex <font color="Olive" title="b1">P</font> being   <a href="topalg_1.html#NM1" title="TOPALG_1:NM.1">Loop</a> of <font color="Maroon" title="c2">t</font> st <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="topalg_1.html#K4" title="TOPALG_1:func.4">EqRel</a> (<font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">t</font>)</span>)</span>,<font color="Olive" title="b1">P</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_1.html#T47" onmouseover="rs('topalg_1/T47')" onmouseout="rh()">TOPALG_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:20_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A5</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">thus </span><a NAME="E4:20"/>
<font color="Maroon" title="c1">T</font> is  <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">pathwise_connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A5</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>
