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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">S</font> be  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">S</font>;<br/>





<b>assume </b><a NAME="E1:55"/>
 <a href="waybel26.html#K5" title="WAYBEL26:func.5">*graph</a> <font color="Maroon" title="c4">f</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">AA</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E3:55"/><i><font color="Green" title="E43">A1</font></i>: 
 <a href="waybel26.html#K5" title="WAYBEL26:func.5">*graph</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">AA</font>
 <b>and </b><br/><a NAME="E4:55"/><i><font color="Green" title="E44">A2</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AA</font> holds <br/> ex <font color="Olive" title="b2">X1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> &amp; <font color="Olive" title="b2">X1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">Y1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T45" title="BORSUK_1:th.45">BORSUK_1:45</a></i>;<br/>
<a NAME="E5:55"/><i><font color="Green" title="E45">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E6:55"/><i><font color="Green" title="E46">A4</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E7:55"/><i><font color="Green" title="E47">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S</font>;<br/><b>assume </b><a NAME="E1:55_1"/><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c6">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:55_1_1_1"/><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P</font>
 ;<br/><a NAME="E2:55_1_1_1"/><b>then </b><i><font color="Green" title="E50">A8</font></i>: 
( <font color="Maroon" title="c7">x</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 <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="waybel26.html#E1:55_1_1_1"><i><font color="Green" title="E49">A7</font></i></a></i>;<br/><a NAME="E4:55_1_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E5:55"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="waybel26.html#E6:55"><i><font color="Green" title="E46">A4</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">W</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">$2 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default">$2 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel26.html#K5" title="WAYBEL26:func.5">*graph</a> <font color="Maroon" title="c4">f</font> );<br/><div><i><font color="Green" title="E51">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:55_1_1_1_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 ;<br/><a NAME="E2:55_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel26.html#K5" title="WAYBEL26:func.5">*graph</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E7:55"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="waybel26.html#E1:55_1_1_1"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="waybel26.html#T39" target="_self" title="WAYBEL26:th.39">Th39</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">V</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:55_1_1_1_1"/><i><font color="Green" title="E52">A10</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c10">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> &amp; <font color="Maroon" title="c11">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AA</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E3:55"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c13">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E6:55_1_1_1_1"/><i><font color="Green" title="E53">A11</font></i>: 
( <font color="Maroon" title="c11">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> &amp; <font color="Maroon" title="c12">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c13">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="waybel26.html#E4:55"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="waybel26.html#E4:55_1_1_1_1"><i><font color="Green" title="E52">A10</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">u</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c15">u</font> = <font color="Maroon" title="c14">u</font>;<br/><a NAME="E8:55_1_1_1_1"/>
( <font color="Maroon" title="c12">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c13">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E6:55_1_1_1_1"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E9:55_1_1_1_1"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default">the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:55_1_1_1_1"/>
( <font color="Maroon" title="c12">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">u</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c13">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">u</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E11:55_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">e</font>,<font color="Maroon" title="c15">u</font>]
 <b>by </b><i><a class="txt" href="waybel26.html#E3:55"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="waybel26.html#E4:55_1_1_1_1"><i><font color="Green" title="E52">A10</font></i></a>, <a class="txt" href="waybel26.html#E6:55_1_1_1_1"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c10">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:55_1_1_1"/><i><font color="Green" title="E52">A12</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default">the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )
 <b>and </b><br/><a NAME="E9:55_1_1_1"/><i><font color="Green" title="E53">A13</font></i>: 
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> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="waybel26.html#E6:55_1_1_1"><i><font color="Green" title="E51">A9</font></i></a>);<br/></i><a NAME="E10:55_1_1_1"/><i><font color="Green" title="E54">A14</font></i>: 
 <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E8:55_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="funct_5.html#T13" title="FUNCT_5:th.13">FUNCT_5:13</a></i>;<br/><a NAME="E11:55_1_1_1"/><i><font color="Green" title="E55">A15</font></i>: 
 <a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E8:55_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="funct_5.html#T13" title="FUNCT_5:th.13">FUNCT_5:13</a></i>;<br/><a NAME="E12:55_1_1_1"/><i><font color="Green" title="E56">A16</font></i>: 
 <a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_1_1_2"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:55_1_1_1_2"/><i><font color="Green" title="E57">A17</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z1</font>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font>
 <b>by </b><i><a class="ref" href="funct_5.html#D2" title="FUNCT_5:def.2">FUNCT_5:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:55_1_1_1_2"/><i><font color="Green" title="E58">A18</font></i>: 
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">g</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z1</font>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E3:55_1_1_1_2"><i><font color="Green" title="E57">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:55_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z1</font>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">z1</font>,<font color="Maroon" title="c11">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E7:55_1_1_1_2"/><b>then </b><i><font color="Green" title="E59">A19</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">z1</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c12">z1</font>,<font color="Maroon" title="c11">z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel26.html#K5" title="WAYBEL26:func.5">*graph</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="txt" href="waybel26.html#E8:55_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="waybel26.html#E9:55_1_1_1"><i><font color="Green" title="E53">A13</font></i></a>, <a class="txt" href="waybel26.html#E5:55_1_1_1_2"><i><font color="Green" title="E58">A18</font></i></a></i>;<br/>
<a NAME="E8:55_1_1_1_2"/>
<font color="Maroon" title="c11">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_1_1_2_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">z</font>
 ;<br/>

<a NAME="E2:55_1_1_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c14">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c12">z1</font>,<font color="Maroon" title="c11">z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel26.html#E7:55_1_1_1_2"><i><font color="Green" title="E59">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E3:55_1_1_1_2_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E7:55_1_1_1_2"><i><font color="Green" title="E59">A19</font></i></a>, <a class="ref" href="waybel26.html#T39" target="_self" title="WAYBEL26:th.39">Th39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:55_1_1_1_2"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c11">J</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> : <font color="Olive" title="b1">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c12">A0</font> being  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>;<br/><a NAME="E14:55_1_1_1"/><i><font color="Green" title="E57">A20</font></i>: 
<font color="Maroon" title="c12">A0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> : <font color="Olive" title="b1">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/><a NAME="E15:55_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> : <font color="Olive" title="b1">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_1_1_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> : <font color="Olive" title="b1">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:55_1_1_1_3"/><i><font color="Green" title="E58">A21</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Maroon" title="c14">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/>
<a NAME="E4:55_1_1_1_3"/><i><font color="Green" title="E59">A22</font></i>: 
<font color="Maroon" title="c14">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="waybel26.html#E11:55_1_1_1"><i><font color="Green" title="E55">A15</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:55_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c14">A</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:55_1_1_1_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E3:55_1_1_1_3"><i><font color="Green" title="E58">A21</font></i></a>, <a class="txt" href="waybel26.html#E4:55_1_1_1_3"><i><font color="Green" title="E59">A22</font></i></a>, <a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c13">J</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> : <font color="Olive" title="b1">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="txt" href="waybel26.html#E14:55_1_1_1"><i><font color="Green" title="E57">A20</font></i></a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><a NAME="E17:55_1_1_1"/>
<font color="Maroon" title="c13">J</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E1:55_1_1_1_4"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:55_1_1_1_4"/><i><font color="Green" title="E58">A23</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c16">A</font> &amp; <font color="Maroon" title="c16">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/>
<b>assume </b><a NAME="E4:55_1_1_1_4"/>
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:55_1_1_1_4"/><i><font color="Green" title="E59">A24</font></i>: 
( <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c17">B</font> &amp; <font color="Maroon" title="c17">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/>
<b>take </b><font color="Maroon" title="c18">ab</font> = <font color="Maroon" title="c14">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c15">b</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c19">AB</font> = <font color="Maroon" title="c16">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c17">B</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel26.html#E3:55_1_1_1_4"><i><font color="Green" title="E58">A23</font></i></a>, <a class="txt" href="waybel26.html#E6:55_1_1_1_4"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="finset_1.html#T14" title="FINSET_1:th.14">FINSET_1:14</a></i>;<br/>
<a NAME="E8:55_1_1_1_4"/><i><font color="Green" title="E60">A25</font></i>: 
(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c19">AB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">b</font> &amp; <font color="Maroon" title="c14">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">ab</font> )
 
<b>by </b><i><a class="txt" href="waybel26.html#E3:55_1_1_1_4"><i><font color="Green" title="E58">A23</font></i></a>, <a class="txt" href="waybel26.html#E6:55_1_1_1_4"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="waybel14.html#T18" title="WAYBEL14:th.18">WAYBEL14:18</a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></i>;<br/>
<b>hence </b><a NAME="E9:55_1_1_1_4"/>
<font color="Maroon" title="c18">ab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J</font>
 ;<br/>

<a NAME="E10:55_1_1_1_4"/>
( <font color="Maroon" title="c14">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">ab</font> &amp; <font color="Maroon" title="c15">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">ab</font> )
 
<b>by </b><i><a class="txt" href="waybel26.html#E8:55_1_1_1_4"><i><font color="Green" title="E60">A25</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E11:55_1_1_1_4"/>
( <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c18">ab</font> &amp; <font color="Maroon" title="c15">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c18">ab</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c14">J'</font> = <font color="Maroon" title="c13">J</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S</font> <b>by </b><i><a class="txt" href="waybel26.html#E6:55"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="waybel_0.html#T3" title="WAYBEL_0:th.3">WAYBEL_0:3</a></i>;<br/><a NAME="E19:55_1_1_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c13">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:55_1_1_1_5"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c13">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_1_1_5_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c13">J</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c16">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:55_1_1_1_5_1"/><i><font color="Green" title="E58">A26</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">u</font> &amp; <font color="Maroon" title="c16">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:55_1_1_1_5_1"/><i><font color="Green" title="E59">A27</font></i>: 
( <font color="Maroon" title="c16">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c17">A</font> &amp; <font color="Maroon" title="c17">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="waybel26.html#E3:55_1_1_1_5_1"><i><font color="Green" title="E58">A26</font></i></a></i>;<br/>
<a NAME="E6:55_1_1_1_5_1"/>
<font color="Maroon" title="c17">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel26.html#E12:55_1_1_1"><i><font color="Green" title="E56">A16</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:55_1_1_1_5_1"/><b>then </b>
<font color="Maroon" title="c16">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel26.html#E5:55_1_1_1_5_1"><i><font color="Green" title="E59">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E8:55_1_1_1_5_1"/><b>then </b>
<font color="Maroon" title="c16">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T99" title="ZFMISC_1:th.99">ZFMISC_1:99</a></i>;<br/>
<b>hence </b><a NAME="E9:55_1_1_1_5_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E3:55_1_1_1_5_1"><i><font color="Green" title="E58">A26</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c15">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:55_1_1_1_5"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 ;<br/>

<a NAME="E3:55_1_1_1_5"/><b>then </b><i><font color="Green" title="E58">A28</font></i>: 
( <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font> &amp; <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 
<b>by </b><i><a class="txt" href="waybel26.html#E8:55_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="waybel26.html#E9:55_1_1_1"><i><font color="Green" title="E53">A13</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:55_1_1_1_5"/><b>then </b>
<font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel26.html#E8:55_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E5:55_1_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel26.html#E3:55_1_1_1_5"><i><font color="Green" title="E58">A28</font></i></a>, <a class="ref" href="funct_5.html#D2" title="FUNCT_5:def.2">FUNCT_5:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">A</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:55_1_1_1_5"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c16">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>;<br/>
<a NAME="E8:55_1_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J</font>
 
;<br/>
<b>hence </b><a NAME="E9:55_1_1_1_5"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c13">J</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E3:55_1_1_1_5"><i><font color="Green" title="E58">A28</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:55_1_1_1"/><b>then </b>
(  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c13">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c13">J</font>,<font color="Maroon" title="c3">S</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a>, <a class="ref" href="yellow_1.html#T22" title="YELLOW_1:th.22">YELLOW_1:22</a></i>;<br/><a NAME="E21:55_1_1_1"/><b>then </b>
(  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c14">J'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c6">P</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> )
 <b>by </b><i><a class="txt" href="waybel26.html#E6:55"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="waybel26.html#E1:55_1"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a>, <a class="ref" href="yellow_0.html#T26" title="YELLOW_0:th.26">YELLOW_0:26</a></i>;<br/><a NAME="E22:55_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">J</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">P</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E2:55_1_1_1"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="waybel11.html#D1" title="WAYBEL11:def.1">WAYBEL11:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E24:55_1_1_1"/><i><font color="Green" title="E58">A29</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J</font> &amp; <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c15">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S</font> <b>by </b><i><a class="txt" href="waybel26.html#E24:55_1_1_1"><i><font color="Green" title="E58">A29</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E27:55_1_1_1"/><i><font color="Green" title="E59">A30</font></i>: 
<font color="Maroon" title="c16">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c17">A</font>
 <b>and </b><br/><a NAME="E28:55_1_1_1"/><i><font color="Green" title="E60">A31</font></i>: 
<font color="Maroon" title="c17">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="waybel26.html#E24:55_1_1_1"><i><font color="Green" title="E58">A29</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">c1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">c1</font>,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">c1</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">c1</font>,$1</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel26.html#K5" title="WAYBEL26:func.5">*graph</a> <font color="Maroon" title="c4">f</font> );<br/><div><i><font color="Green" title="E61">A32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:55_1_1_1_6"/>
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">A</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c19">c1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:55_1_1_1_6"/><i><font color="Green" title="E62">A33</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">c1</font>,<font color="Maroon" title="c18">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font>
 <b>by </b><i><a class="ref" href="funct_5.html#D2" title="FUNCT_5:def.2">FUNCT_5:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:55_1_1_1_6"/><i><font color="Green" title="E63">A34</font></i>: 
( <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">g</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">c1</font>,<font color="Maroon" title="c18">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">a</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E3:55_1_1_1_6"><i><font color="Green" title="E62">A33</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>take </b><font color="Maroon" title="c21">a</font> = <font color="Maroon" title="c20">a</font>;<br/><b>thus </b><a NAME="E6:55_1_1_1_6"/>
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">W</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E8:55_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="waybel26.html#E5:55_1_1_1_6"><i><font color="Green" title="E63">A34</font></i></a></i>;<br/><a NAME="E7:55_1_1_1_6"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">c1</font>,<font color="Maroon" title="c18">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">c1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">c1</font>,<font color="Maroon" title="c18">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">c</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><a NAME="E8:55_1_1_1_6"/><b>then </b>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">c1</font> &amp; <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">c</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c19">c1</font>,<font color="Maroon" title="c18">c</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel26.html#K5" title="WAYBEL26:func.5">*graph</a> <font color="Maroon" title="c4">f</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E8:55_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="waybel26.html#E9:55_1_1_1"><i><font color="Green" title="E53">A13</font></i></a>, <a class="txt" href="waybel26.html#E5:55_1_1_1_6"><i><font color="Green" title="E63">A34</font></i></a></i>;<br/><b>hence </b><a NAME="E9:55_1_1_1_6"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">c</font>,<font color="Maroon" title="c21">a</font>]
 <b>by </b><i><a class="txt" href="waybel26.html#E8:55_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="waybel26.html#E5:55_1_1_1_6"><i><font color="Green" title="E63">A34</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c18">hh</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E31:55_1_1_1"/><i><font color="Green" title="E62">A35</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">hh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">A</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">W</font> )
 <b>and </b><br/><a NAME="E32:55_1_1_1"/><i><font color="Green" title="E63">A36</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">A</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c18">hh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="waybel26.html#E29:55_1_1_1"><i><font color="Green" title="E61">A32</font></i></a>);<br/></i><b>set </b><font color="Maroon" title="c19">B</font> =  <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font></span>)</span></span>)</span>;<br/><a NAME="E33:55_1_1_1"/>
<font color="Maroon" title="c10">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></i>;<br/><a NAME="E34:55_1_1_1"/><b>then </b>
 <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_5.html#T5" title="FUNCT_5:th.5">FUNCT_5:5</a></i>;<br/><a NAME="E35:55_1_1_1"/><b>then </b><i><font color="Green" title="E64">A37</font></i>: 
 <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E10:55_1_1_1"><i><font color="Green" title="E54">A14</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E36:55_1_1_1"/><b>then </b>
 <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">B</font> =  <a href="funct_5.html#K1" title="FUNCT_5:func.1">proj1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/><b>reconsider </b><font color="Maroon" title="c21">B</font> = <font color="Maroon" title="c20">B</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/><b>reconsider </b><font color="Maroon" title="c22">Q</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c21">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><b>take </b><font color="Maroon" title="c23">Q</font> = <font color="Maroon" title="c22">Q</font>;<br/><a NAME="E40:55_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="waybel26.html#E28:55_1_1_1"><i><font color="Green" title="E60">A31</font></i></a>, <a class="txt" href="waybel26.html#E31:55_1_1_1"><i><font color="Green" title="E62">A35</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/><a NAME="E41:55_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T17" title="FINSET_1:th.17">FINSET_1:17</a></i>;<br/><a NAME="E42:55_1_1_1"/><b>then </b>
<font color="Maroon" title="c21">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="waybel26.html#T40" target="_self" title="WAYBEL26:th.40">Th40</a></i>;<br/><a NAME="E43:55_1_1_1"/><b>then </b>
<font color="Maroon" title="c23">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E35:55_1_1_1"><i><font color="Green" title="E64">A37</font></i></a>, <a class="ref" href="cantor_1.html#D4" title="CANTOR_1:def.4">CANTOR_1:def 4</a></i>;<br/><a NAME="E44:55_1_1_1"/><b>then </b>
<font color="Maroon" title="c23">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#T5" title="CANTOR_1:th.5">CANTOR_1:5</a></i>;<br/><b>hence </b><a NAME="E45:55_1_1_1"/>
<font color="Maroon" title="c23">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><b>thus </b><a NAME="E46:55_1_1_1"/>
<font color="Maroon" title="c23">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c24">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_1_1_7"/><i><font color="Green" title="E65">A38</font></i>: 
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Q</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c25">zz</font> = <font color="Maroon" title="c24">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c26">fz</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">zz</font>, <font color="Maroon" title="c27">aa</font> = <font color="Maroon" title="c16">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="txt" href="waybel26.html#E6:55"><i><font color="Green" title="E46">A4</font></i></a></i>;<br/>
<a NAME="E4:55_1_1_1_7"/>
<font color="Maroon" title="c16">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">zz</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c28">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_1_1_7_1"/>
<font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c29">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:55_1_1_1_7_1"/><i><font color="Green" title="E66">A39</font></i>: 
( <font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">q</font> &amp; <font color="Maroon" title="c29">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">A</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E27:55_1_1_1"><i><font color="Green" title="E59">A30</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c30">q1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:55_1_1_1_7_1"/><i><font color="Green" title="E67">A40</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">q1</font>,<font color="Maroon" title="c29">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">hh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">q</font></span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">q1</font> &amp; <font color="Maroon" title="c18">hh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">q</font> &amp; <font color="Maroon" title="c18">hh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c30">q1</font>,<font color="Maroon" title="c29">q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel26.html#K5" title="WAYBEL26:func.5">*graph</a> <font color="Maroon" title="c4">f</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E32:55_1_1_1"><i><font color="Green" title="E63">A36</font></i></a>, <a class="txt" href="waybel26.html#E3:55_1_1_1_7_1"><i><font color="Green" title="E66">A39</font></i></a></i>;<br/>
<a NAME="E6:55_1_1_1_7_1"/>
<font color="Maroon" title="c18">hh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font>
 
<b>by </b><i><a class="txt" href="waybel26.html#E31:55_1_1_1"><i><font color="Green" title="E62">A35</font></i></a>, <a class="txt" href="waybel26.html#E3:55_1_1_1_7_1"><i><font color="Green" title="E66">A39</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:55_1_1_1_7_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">q1</font>,<font color="Maroon" title="c29">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel26.html#E8:55_1_1_1"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="waybel26.html#E5:55_1_1_1_7_1"><i><font color="Green" title="E67">A40</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E8:55_1_1_1_7_1"/><b>then </b>
<font color="Maroon" title="c30">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">B</font>
 
<b>by </b><i><a class="ref" href="funct_5.html#D1" title="FUNCT_5:def.1">FUNCT_5:def 1</a></i>;<br/>
<a NAME="E9:55_1_1_1_7_1"/><b>then </b>
<font color="Maroon" title="c25">zz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c30">q1</font>
 
<b>by </b><i><a class="txt" href="waybel26.html#E1:55_1_1_1_7"><i><font color="Green" title="E65">A38</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/>
<a NAME="E10:55_1_1_1_7_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">zz</font>,<font color="Maroon" title="c28">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c30">q1</font>,<font color="Maroon" title="c29">q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel26.html#E3:55_1_1_1_7_1"><i><font color="Green" title="E66">A39</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E11:55_1_1_1_7_1"/>
<font color="Maroon" title="c28">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">zz</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E5:55_1_1_1_7_1"><i><font color="Green" title="E67">A40</font></i></a>, <a class="ref" href="waybel26.html#T39" target="_self" title="WAYBEL26:th.39">Th39</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:55_1_1_1_7"/><b>then </b>
<font color="Maroon" title="c27">aa</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c26">fz</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<a NAME="E6:55_1_1_1_7"/><b>then </b>
( <font color="Maroon" title="c16">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">zz</font> &amp; <font color="Maroon" title="c6">P</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 
<b>by </b><i><a class="txt" href="waybel26.html#E6:55"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="waybel26.html#E1:55_1"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<a NAME="E7:55_1_1_1_7"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">zz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
<b>by </b><i><a class="txt" href="waybel26.html#E24:55_1_1_1"><i><font color="Green" title="E58">A29</font></i></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></i>;<br/>
<b>hence </b><a NAME="E8:55_1_1_1_7"/>
<font color="Maroon" title="c24">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">c1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:55_1_1_1_8"/>
<font color="Maroon" title="c24">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">B</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c25">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:55_1_1_1_8"/><i><font color="Green" title="E65">A41</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">c1</font>,<font color="Maroon" title="c25">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">g</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_5.html#D1" title="FUNCT_5:def.1">FUNCT_5:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c26">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:55_1_1_1_8"/><i><font color="Green" title="E66">A42</font></i>: 
( <font color="Maroon" title="c26">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">g</font> &amp; <font color="Maroon" title="c26">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">hh</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">c1</font>,<font color="Maroon" title="c25">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">b</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E3:55_1_1_1_8"><i><font color="Green" title="E65">A41</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>consider </b><font color="Maroon" title="c27">c'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:55_1_1_1_8"/><i><font color="Green" title="E67">A43</font></i>: 
( <font color="Maroon" title="c27">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">hh</font> &amp; <font color="Maroon" title="c26">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">hh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">c'</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E5:55_1_1_1_8"><i><font color="Green" title="E66">A42</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c28">c'1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:55_1_1_1_8"/><i><font color="Green" title="E68">A44</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">c'1</font>,<font color="Maroon" title="c27">c'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">hh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">c'</font></span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">c'1</font> &amp; <font color="Maroon" title="c18">hh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">c'</font> &amp; <font color="Maroon" title="c18">hh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c28">c'1</font>,<font color="Maroon" title="c27">c'</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel26.html#K5" title="WAYBEL26:func.5">*graph</a> <font color="Maroon" title="c4">f</font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E31:55_1_1_1"><i><font color="Green" title="E62">A35</font></i></a>, <a class="txt" href="waybel26.html#E32:55_1_1_1"><i><font color="Green" title="E63">A36</font></i></a>, <a class="txt" href="waybel26.html#E7:55_1_1_1_8"><i><font color="Green" title="E67">A43</font></i></a></i>;<br/><b>thus </b><a NAME="E10:55_1_1_1_8"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">c1</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E5:55_1_1_1_8"><i><font color="Green" title="E66">A42</font></i></a>, <a class="txt" href="waybel26.html#E7:55_1_1_1_8"><i><font color="Green" title="E67">A43</font></i></a>, <a class="txt" href="waybel26.html#E9:55_1_1_1_8"><i><font color="Green" title="E68">A44</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E48:55_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Q</font>
 <b>by </b><i><a class="txt" href="waybel26.html#E1:55_1_1_1"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:55_1_1"/>
 ex <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> )
 ;<br/><b>hence </b><a NAME="E3:55_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:55_1"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57" title="TOPS_1:th.57">TOPS_1:57</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:55"/>
<font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


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