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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">Affn</font> being   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Olive" title="b1">n</font></span>)</span>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">Affn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 holds <br/> <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Olive" title="b2">Affn</font>,<font color="Maroon" title="c1">x</font>) is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Affn</font> being   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">Affn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 holds <br/> <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Olive" title="b1">Affn</font>,<font color="Maroon" title="c1">x</font>) is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">Affn</font> be   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">Affn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies  <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>) is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">TRn</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">AA</font> =  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">Affn</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">Ax</font> =  <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">AxA</font> = <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">Affn</font></span>)</span> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">Affn</font></span>)</span></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#T31" target="_self" onmouseover="rs('rlaffin3/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:50"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">Affn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>) is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/>

<a NAME="E3:50"/>
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#T4" target="_self" onmouseover="rs('rlaffin3/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E4:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A2</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">Affn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="rlaffin3.html#T6" target="_self" onmouseover="rs('rlaffin3/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E5:50"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">Affn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T93" onmouseover="rs('tsep_1/T93')" onmouseout="rh()">TSEP_1:93</a></span>;<br/>
<a NAME="E6:50"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">Affn</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Olive" title="b1">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  holds <br/><span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Olive" title="b1">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  implies <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:50_1"/>
<font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/><a NAME="E2:50_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">AxA</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D6" onmouseover="rs('pre_topc/D6')" onmouseout="rh()">PRE_TOPC:def 6</a></span>;<br/><a NAME="E3:50_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">AxA</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/><a NAME="E4:50_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">AxA</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="relat_1.html#T69" onmouseover="rs('relat_1/T69')" onmouseout="rh()">RELAT_1:69</a></span>;<br/><a NAME="E5:50_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_1"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:50_1"/>
<span class="p1">(<span class="default"><a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>)</span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c8">P</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T3" onmouseover="rs('tops_1/T3')" onmouseout="rh()">TOPS_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:50"/>
 <a href="rlaffin3.html#K2" title="RLAFFIN3:func.2">|--</a> (<font color="Maroon" title="c3">Affn</font>,<font color="Maroon" title="c1">x</font>) is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D6" onmouseover="rs('pre_topc/D6')" onmouseout="rh()">PRE_TOPC:def 6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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