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<span class="kw">let </span><font color="Maroon" title="c1">X</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">Y</font> being   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 <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b4">y1</font>, <font color="Olive" title="b5">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">Y</font><br/>  for <font color="Olive" title="b6">f</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b4">y1</font>,<font color="Olive" title="b5">y2</font>  st <font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y1</font> &amp; <font color="Olive" title="b3">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">y2</font> &amp; <font color="Olive" title="b4">y1</font>,<font color="Olive" title="b5">y2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/><font color="Olive" title="b6">f</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b2">x1</font>,<font color="Olive" title="b3">x2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</font> be   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 <font color="Maroon" title="c1">X</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">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font><br/>  for <font color="Olive" title="b5">f</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b3">y1</font>,<font color="Olive" title="b4">y2</font>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y1</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y2</font> &amp; <font color="Olive" title="b3">y1</font>,<font color="Olive" title="b4">y2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/><font color="Olive" title="b5">f</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</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">y1</font>, <font color="Olive" title="b2">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font><br/>  for <font color="Olive" title="b3">f</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Olive" title="b1">y1</font>,<font color="Olive" title="b2">y2</font>  st <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y1</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y2</font> &amp; <font color="Olive" title="b1">y1</font>,<font color="Olive" title="b2">y2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/><font color="Olive" title="b3">f</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">y1</font>, <font color="Maroon" title="c6">y2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</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">f</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>  st <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> &amp; <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  holds <br/><font color="Olive" title="b1">f</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">f</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> &amp; <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  implies <font color="Maroon" title="c7">f</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:5"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
( <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> )
 <span class="kw">and </span><br/><a NAME="E2:5"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</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"> <font color="Maroon" title="c7">f</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><br/>

<a NAME="E3:5"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" onmouseover="rs('tsep_1/T1')" onmouseout="rh()">TSEP_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>,<font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" onmouseover="rs('funct_2/T7')" onmouseout="rh()">FUNCT_2:7</a></span>;<br/>
<a NAME="E5:5"/>
<font color="Maroon" title="c7">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E3">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/>
<a NAME="E6:5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A3</font></span>: 
<font color="Maroon" title="c8">g</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T26" onmouseover="rs('pre_topc/T26')" onmouseout="rh()">PRE_TOPC:26</a></span>;<br/>
<a NAME="E7:5"/><span class="lab"><font color="Green" title="E5">A4</font></span>: 
( <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c3">x1</font> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E3">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/>
<a NAME="E8:5"/><span class="kw">then </span>
<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</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="#E6:5"><span class="lab"><font color="Green" title="E4">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:5"/>
<font color="Maroon" title="c7">f</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:5"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="txt" href="#E6:5"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="borsuk_2.html#D2" onmouseover="rs('borsuk_2/D2')" onmouseout="rh()">BORSUK_2:def 2</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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