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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K7" title="RELSET_1:func.7">.:</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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K7" title="RELSET_1:func.7">.:</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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>
 <span class="kw">and </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:16"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E7:16"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:16"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E9:16"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c4">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E10:16"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c5">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E11:16"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#T10" target="_self" title="CONNSP_1:th.10">Th11</a></span>;<br/>
<a NAME="E12:16"/><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></span>;<br/>
<a NAME="E13:16"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c5">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></span>;<br/>
<a NAME="E14:16"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c5">B</font></span>)</span> = 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="relat_1.html#T136" title="RELAT_1:th.136">RELAT_1:136</a></span>
;<br/>
<a NAME="E16:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A13</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<a NAME="E17:16"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
 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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E28">A15</font></span>:  <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#K8" title="RELSET_1:func.8">"</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">GY</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T40" title="FUNCT_2:th.40">FUNCT_2:40</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c5">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="relat_1.html#T140" title="RELAT_1:th.140">RELAT_1:140</a></span>
;<br/>
<a NAME="E19:16"/><span class="lab"><font color="Green" title="E29">A16</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K7" title="RELSET_1:func.7">.:</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">GX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="funct_2.html#T37" title="FUNCT_2:th.37">FUNCT_2:37</a></span>;<br/>
<a NAME="E20:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="relat_1.html#T139" title="RELAT_1:th.139">RELAT_1:139</a></span>;<br/>
<a NAME="E21:16"/>
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="relat_1.html#T139" title="RELAT_1:th.139">RELAT_1:139</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:16"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E20:16"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="connsp_1.html#T10" target="_self" title="CONNSP_1:th.10">Th11</a></span>; <a class="txt" 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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