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<b>let </b><font color="Maroon" title="c1">GX</font>, <font color="Maroon" title="c2">GY</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">GX</font>,<font color="Maroon" title="c2">GY</font>  st <font color="Olive" title="b1">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></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="c2">GY</font> &amp; <font color="Maroon" title="c1">GX</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> holds <br/><font color="Maroon" title="c2">GY</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a></span><br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">GX</font>,<font color="Maroon" title="c2">GY</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></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="c2">GY</font> &amp; <font color="Maroon" title="c1">GX</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> implies <font color="Maroon" title="c2">GY</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></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="c2">GY</font>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c1">GX</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">GY</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a></span><br/>

<b>assume </b><a NAME="E4:16"/>
not <font color="Maroon" title="c2">GY</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">GY</font><b> such that </b><br/><a NAME="E6:16"/><i><font color="Green" title="E17">A4</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">GY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">B</font>
 <b>and </b><br/><a NAME="E7:16"/><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">GY</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">GY</font> )
 <b>and </b><br/><a NAME="E8:16"/><i><font color="Green" title="E19">A6</font></i>: 
( <font color="Maroon" title="c4">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c5">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>and </b><br/><a NAME="E9:16"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T11" target="_self" title="CONNSP_1:th.11">Th11</a></i>;<br/>
<a NAME="E10:16"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E9:16"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E11:16"/><i><font color="Green" title="E22">A9</font></i>: 
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">GY</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="#E7:16"><i><font color="Green" title="E18">A5</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E23">A10</font></i>:  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font> = 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">GY</font>
<b>by </b><i><a class="ref" href="funct_2.html#T48" title="FUNCT_2:th.48">FUNCT_2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">B</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:16"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="relat_1.html#T175" title="RELAT_1:th.175">RELAT_1:175</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">B</font></span>)</span> = 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="#E10:16"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="relat_1.html#T171" title="RELAT_1:th.171">RELAT_1:171</a></i>
;<br/>
<a NAME="E14:16"/><b>then </b><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E15:16"/><i><font color="Green" title="E25">A12</font></i>: 
( <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="#E8:16"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/>
<a NAME="E16:16"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GX</font>
 
<b>by </b><i><a class="txt" href="#E11:16"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="funct_2.html#T45" title="FUNCT_2:th.45">FUNCT_2:45</a></i>;<br/>
<a NAME="E17:16"/><b>then </b>
( <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> &amp; <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> )
 
<b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E7:16"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="relat_1.html#T174" title="RELAT_1:th.174">RELAT_1:174</a></i>;<br/>
<b>hence </b><a NAME="E18:16"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="#E12:16"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="#E14:16"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="#E15:16"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="connsp_1.html#T11" target="_self" title="CONNSP_1:th.11">Th11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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