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<span class="kw">reconsider </span><font color="Maroon" title="c1">00</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c2">01</font> = 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">w</font> being  <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b2">f</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>for <font color="Olive" title="b3">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  holds <br/> ( ( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( not <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">X</font> =  <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">w</font> be   <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  st <br/>for <font color="Olive" title="b2">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  holds <br/> ( ( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( not <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">V</font> = <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_5.html#T61" target="_self" title="TOPGEN_5:th.61">Th61</a>, <a class="ref" href="topgen_3.html#T11" title="TOPGEN_3:th.11">TOPGEN_3:11</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = 1;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">f1</font> = <span class="p1">(<span class="default"><a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">V</font></span>)</span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c1">00</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">V</font></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">f2</font> = <span class="p1">(<span class="default"><a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="struct_0.html#K6" title="STRUCT_0:func.6">--&gt;</a> <font color="Maroon" title="c2">01</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ;<br/>
<a NAME="E5:68"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw"> such that </span><br/><a NAME="E7:68"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] implies <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] implies <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S5" title="FUNCT_2:sch.5">FUNCT_2:sch 5</a>(<a class="txt" href="#E5:68"><span class="lab"><font color="Green" title="E48">A1</font></span></a>);<br/></span>
<a NAME="E8:68"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<a NAME="E9:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:68"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></span>;<br/>
<a NAME="E11:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E12:68"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E55">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f2</font></span>)</span> implies ( ( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f2</font> implies <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> ) &amp; ( not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f2</font> implies <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> ) ) )</span><br/><span class="kw">assume </span><a NAME="E1:68_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f2</font> implies <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> ) &amp; ( not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f2</font> implies <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> ) )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f2</font> implies <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:68_1_1"/><span class="lab"><font color="Green" title="E56">A9</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font></span><br/><a NAME="E2:68_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A10</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E3:68_1_1"/>
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1_1"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E4:68_1_1"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68_1_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E49">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E4:68_1"/>
not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font></span><br/><a NAME="E5:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E49">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E5:68_1"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:68"/>
<font color="Maroon" title="c6">V</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></span>;<br/>
<a NAME="E15:68"/><span class="kw">then </span>
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c8">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:68"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E12:68"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E11:68"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_5.html#T16" target="_self" title="TOPGEN_5:th.16">Th16</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a>  holds <br/> ( ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topgen_3.html#K2" title="TOPGEN_3:func.2">Sorgenfrey-line</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( not <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<span class="kw">thus </span><a NAME="E17:68"/>
( ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( not <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">w</font></span><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">.[</a></span> implies <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E49">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
