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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">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><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="b1">t1</font>,<font color="Olive" title="b2">t2</font>  st <font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a>  holds <br/><font color="Olive" title="b3">p</font> is   <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></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">  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="c2">t1</font>,<font color="Maroon" title="c3">t2</font>  st <font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a>  holds <br/><font color="Olive" title="b1">p</font> is   <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></span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</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"> ( <font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</font> <a href="borsuk_2.html#R1" title="BORSUK_2:pred.1">are_connected</a>  implies <font color="Maroon" title="c4">p</font> is   <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> )</span><br/>





<span class="kw">assume </span><a NAME="E1:54"/>
<font color="Maroon" title="c2">t1</font>,<font color="Maroon" title="c3">t2</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="c4">p</font> is   <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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">c</font> = <font color="Maroon" title="c4">p</font> as   <a href="topalg_6.html#NM1" title="TOPALG_6:NM.1">Curve</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="topalg_6.html#T22" target="_self" onmouseover="rs('topalg_6/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<a NAME="E3:54"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">c</font>
 
<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="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:54"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E5:54"/><span class="kw">then </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="c5">c</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E29">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="E6:54"/><span class="kw">then </span>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">c</font> is  <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D5" onmouseover="rs('xxreal_2/D5')" onmouseout="rh()">XXREAL_2:def 5</a></span>;<br/>
<a NAME="E7:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c5">c</font> is  <a href="topalg_6.html#V5" title="TOPALG_6:attr.5">with_first_point</a> 
 
;<br/>
<a NAME="E8:54"/>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" onmouseover="rs('xxreal_1/T1')" onmouseout="rh()">XXREAL_1:1</a></span>;<br/>
<a NAME="E9:54"/><span class="kw">then </span>
 <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="c5">c</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E29">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="E10:54"/><span class="kw">then </span>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">c</font> is  <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D6" onmouseover="rs('xxreal_2/D6')" onmouseout="rh()">XXREAL_2:def 6</a></span>;<br/>
<a NAME="E11:54"/><span class="kw">then </span>
<font color="Maroon" title="c5">c</font> is  <a href="topalg_6.html#V6" title="TOPALG_6:attr.6">with_last_point</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E12:54"/>
<font color="Maroon" title="c4">p</font> is   <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E30">A2</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/>


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