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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#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> holds   <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b2">f</font> is  <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 <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> holds   <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b1">f</font> is  <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 <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font> is  <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 <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span><br/>





<a NAME="E1:44"/><i><font color="Green" title="E36">A1</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> <font color="Maroon" title="c3">M</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:44"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="waybel26.html#T33" target="_self" title="WAYBEL26:th.33">Th33</a></i>;<br/>
<a NAME="E3:44"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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/>
<a NAME="E4:44"/><b>then </b><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E5:44"/><b>then </b>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T85" title="FUNCT_6:th.85">FUNCT_6:85</a></i>;<br/>
<a NAME="E6:44"/><b>then </b><i><font color="Green" title="E38">A3</font></i>: 
 ex <font color="Olive" title="b1">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</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> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E7:44"/><i><font color="Green" title="E39">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel18.html#D3" title="WAYBEL18:def.3">WAYBEL18:def 3</a></i>;<br/>
<a NAME="E8:44"/><i><font color="Green" title="E40">A5</font></i>: 
 <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="waybel26.html#T31" target="_self" title="WAYBEL26:th.31">Th31</a></i>;<br/>
<a NAME="E9:44"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="#E7:44"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="card_3.html#T20" title="CARD_3:th.20">CARD_3:20</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">g</font> =  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E6:44"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">B</font> =  <a href="waybel18.html#K2" title="WAYBEL18:func.2">product_prebasis</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> as    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c3">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="ref" href="waybel18.html#D3" title="WAYBEL18:def.3">WAYBEL18:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> implies <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/><b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon" title="c7">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/><b>then consider </b><font color="Maroon" title="c8">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c9">T</font> being    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> , <font color="Maroon" title="c10">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">T</font><b> such that </b><br/><a NAME="E3:44_1"/><i><font color="Green" title="E41">A6</font></i>: 
( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c10">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c9">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )
 <b>and </b><br/><a NAME="E4:44_1"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c10">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel18.html#D2" title="WAYBEL18:def.2">WAYBEL18:def 2</a></i>;<br/><a NAME="E5:44_1"/><i><font color="Green" title="E43">A8</font></i>: 
<font color="Maroon" title="c9">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="#E3:44_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">i</font> = <font color="Maroon" title="c8">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> <b>by </b><i><a class="txt" href="#E3:44_1"><i><font color="Green" title="E41">A6</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c12">FP</font> = <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">V</font>;<br/><a NAME="E7:44_1"/><i><font color="Green" title="E44">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">V</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/><a NAME="E8:44_1"/><i><font color="Green" title="E45">A10</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="#E8:44"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E9:44_1"/><b>then </b><i><font color="Green" title="E46">A11</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">V</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">fi</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</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 <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="waybel26.html#T2" target="_self" title="WAYBEL26:th.2">Th2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> implies  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="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> ) ) &amp; (  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="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> ) implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  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="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> ) implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:44_1_1_1"/><i><font color="Green" title="E47">A12</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> )</span><br/><a NAME="E2:44_1_1_1"/><b>then </b>
( <font color="Maroon" title="c14">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="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">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>then consider </b><font color="Maroon" title="c15">gx</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:44_1_1_1"/><i><font color="Green" title="E48">A13</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">gx</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">gx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">V</font></span>)</span> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">V</font></span>)</span> holds <br/><font color="Maroon" title="c15">gx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> ) )
 <b>by </b><i><a class="txt" href="#E4:44_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">Q</font> = <font color="Maroon" title="c13">fi</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">V</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="c17">Q</font> = <font color="Maroon" title="c16">Q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c17">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Q</font> )</span><br/><a NAME="E6:44_1_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>hence </b><a NAME="E7:44_1_1_1"/>
<font color="Maroon" title="c17">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E3:44_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="#E5:44_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Q</font> )</span><br/><b>thus </b><a NAME="E8:44_1_1_1"/>
<font color="Maroon" title="c17">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Q</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Q</font> or <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> )</span><br/>

<b>assume </b><a NAME="E1:44_1_1_1_1"/><i><font color="Green" title="E49">A14</font></i>: 
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font></span><br/>

<a NAME="E2:44_1_1_1_1"/><b>then </b><i><font color="Green" title="E50">A15</font></i>: 
( <font color="Maroon" title="c18">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 <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c13">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">V</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<a NAME="E3:44_1_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g</font>
 
<b>by </b><i><a class="txt" href="#E6:44"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="#E1:44_1_1_1_1"><i><font color="Green" title="E49">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">ga</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:44_1_1_1_1"/><i><font color="Green" title="E51">A16</font></i>: 
( <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ga</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">ga</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">ga</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="txt" href="#E6:44"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c19">ga</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> )</span><br/><b>assume </b><a NAME="E1:44_1_1_1_1_1"/><i><font color="Green" title="E52">A17</font></i>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">ga</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font></span><br/><a NAME="E2:44_1_1_1_1_1"/><b>then </b>
( ( <font color="Maroon" title="c20">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</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="c2">Y</font> ) or <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E3:44_1_1_1_1_1"/><b>then </b>
( ( <font color="Maroon" title="c20">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c19">ga</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</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="c19">ga</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</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="c2">Y</font> ) or <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> )
 <b>by </b><i><a class="txt" href="#E8:44"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="#E5:44_1_1_1_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="txt" href="#E1:44_1_1_1_1_1"><i><font color="Green" title="E52">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E4:44_1_1_1_1_1"/>
<font color="Maroon" title="c19">ga</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</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="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>
 <b>by </b><i><a class="txt" href="#E4:44"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="#E9:44_1"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="#E2:44_1_1_1_1"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="#E5:44_1_1_1_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="funct_6.html#T86" title="FUNCT_6:th.86">FUNCT_6:86</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:44_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c19">ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font>
 
<b>by </b><i><a class="txt" href="#E4:44_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="#E7:44_1"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="#E8:44_1"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="#E5:44_1_1_1_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>hence </b><a NAME="E8:44_1_1_1_1"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font>
 <b>by </b><i><a class="txt" href="#E1:44_1_1_1_1"><i><font color="Green" title="E49">A14</font></i></a>, <a class="txt" href="#E5:44_1_1_1_1"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E9:44_1_1_1"/><i><font color="Green" title="E49">A18</font></i>: 
<font color="Maroon" title="c15">gx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">V</font>
 <b>by </b><i><a class="txt" href="#E7:44_1"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="#E8:44_1"><i><font color="Green" title="E45">A10</font></i></a>, <a class="txt" href="#E9:44_1"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="#E4:44_1_1_1"><i><font color="Green" title="E48">A13</font></i></a></i>;<br/><a NAME="E10:44_1_1_1"/>
<font color="Maroon" title="c15">gx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E4:44"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="#E1:44_1_1_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="#E4:44_1_1_1"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="funct_6.html#T86" title="FUNCT_6:th.86">FUNCT_6:86</a></i>;<br/><b>hence </b><a NAME="E11:44_1_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Q</font>
 <b>by </b><i><a class="txt" href="#E1:44_1_1_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="#E9:44_1_1_1"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>thus </b><a NAME="E2:44_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="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> ) implies <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E12:44_1"/>
<font color="Maroon" title="c5">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:44"/>
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font> is  <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 <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="waybel26.html#NK2" title="WAYBEL26:NK.2">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_9.html#T36" title="YELLOW_9:th.36">YELLOW_9:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
