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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="lopban_1.html#NM2" title="LOPBAN_1:NM.2">RealBanachSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">bounded</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><br/>  for <font color="Olive" title="b2">T1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span>  st <font color="Olive" title="b2">T1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">T</font> &amp; <font color="Olive" title="b2">T1</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  holds <br/><font color="Olive" title="b2">T1</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">T</font> be   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">bounded</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span>  st <font color="Olive" title="b1">T1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Olive" title="b1">T1</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  holds <br/><font color="Olive" title="b1">T1</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">T1</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">T1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c4">T1</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <font color="Maroon" title="c4">T1</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c4">T1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">T</font>
 <span class="kw">and </span><br/><a NAME="E2:17"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c4">T1</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">T1</font> is  <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:17"/>
 for <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/><font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="t_0topsp.html#D2" title="T_0TOPSP:def.2">T_0TOPSP:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c5">TB1</font> = <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,1)</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">TBn</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b1">TBn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,$1)</span>)</span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">TBn</font> );<br/>
<span class="kw">let </span><font color="Maroon" title="c6">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E2:17_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c2">Y</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c2">Y</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">TBn</font> = <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c7">n</font>)</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span> ;<br/>
<a NAME="E2:17_1_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">TBn</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>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:17_1_1"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="c2">Y</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>,<font color="Olive" title="b1">y</font>]
 ; <a class="txt" 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">consider </span><font color="Maroon" title="c7">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><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="c2">Y</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:17_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">f</font> as   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<a NAME="E6:17_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> is  <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> is  <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a> </span><br/>

<a NAME="E1:17_1_2"/>
 ex <font color="Olive" title="b1">TBn</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b1">TBn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c9">n</font>)</span>)</span> &amp; <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">TBn</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:17_1_2"/>
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n</font> is  <a href="nfcont_1.html#V2" title="NFCONT_1:attr.2">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#T15" title="NORMSP_2:th.15">NORMSP_2:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:17_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</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="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">z</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="c2">Y</font> or <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_3"/>
<font color="Maroon" title="c9">z</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="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">z1</font> = <font color="Maroon" title="c9">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<a NAME="E3:17_1_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">T</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 <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E4:17_1_3"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">T</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:17_1_3"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c11">x1</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>
 <span class="kw">and </span><br/><a NAME="E7:17_1_3"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c10">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c11">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_3"><span class="lab"><font color="Green" title="E20">A7</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E10:17_1_3"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c12">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#T8" title="MESFUNC1:th.8">MESFUNC1:8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">n</font> = <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<a NAME="E11:17_1_3"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c12">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:17_1_3"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E12:17_1_3"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c12">x1</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#T2" title="NORMSP_1:th.2">NORMSP_1:2</a></span>;<br/>
<a NAME="E13:17_1_3"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c12">x1</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T14" title="RLVECT_1:th.14">RLVECT_1:14</a></span>;<br/>
<a NAME="E14:17_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
;<br/>
<a NAME="E15:17_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></span>;<br/>
<a NAME="E16:17_1_3"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E17:17_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">TBn</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E19:17_1_3"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c15">TBn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E20:17_1_3"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c15">TBn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a></span>;<br/>
<a NAME="E21:17_1_3"/>
<font color="Maroon" title="c15">TBn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:17_1_3"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:17_1_3"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1_3"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E15:17_1_3"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E19:17_1_3"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E17:17_1_3"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:17_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T11" title="PROB_1:th.11">PROB_1:11</a></span>;<br/>
<a NAME="E9:17_1"/><span class="kw">then </span>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span>)</span> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">n0</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c10">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c11">y0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E11:17_1"/><span class="lab"><font color="Green" title="E20">A14</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 <span class="kw">and </span><br/><a NAME="E12:17_1"/><span class="lab"><font color="Green" title="E21">A15</font></span>: 
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<font color="Maroon" title="c11">y0</font>,<font color="Maroon" title="c10">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="lopban_5.html#T3" title="LOPBAN_5:th.3">LOPBAN_5:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">TBn0</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E14:17_1"/><span class="lab"><font color="Green" title="E22">A16</font></span>: 
<font color="Maroon" title="c12">TBn0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c9">n0</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:17_1"/><span class="lab"><font color="Green" title="E23">A17</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c12">TBn0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">TBn01</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K3" title="NORMSP_2:func.3">TopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E17:17_1"/><span class="lab"><font color="Green" title="E24">A18</font></span>: 
<font color="Maroon" title="c13">TBn01</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E18:17_1"/><span class="lab"><font color="Green" title="E25">A19</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c13">TBn01</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a></span>;<br/>
<a NAME="E19:17_1"/>
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c9">n0</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T14" target="_self" title="LOPBAN_6:th.14">Th14</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E20:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">TBn0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">TBn01</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:17_1"><span class="lab"><font color="Green" title="E22">A16</font></span></a>, <a class="txt" href="#E17:17_1"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="ref" href="relat_1.html#T123" title="RELAT_1:th.123">RELAT_1:123</a></span>;<br/>
<a NAME="E21:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">n0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:17_1"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="txt" href="#E18:17_1"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="ref" href="pre_topc.html#T19" title="PRE_TOPC:th.19">PRE_TOPC:19</a></span>;<br/>
<a NAME="E22:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A20</font></span>: 
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<font color="Maroon" title="c11">y0</font>,<font color="Maroon" title="c10">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c13">TBn01</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:17_1"><span class="lab"><font color="Green" title="E21">A15</font></span></a>, <a class="txt" href="#E18:17_1"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">LTBn01</font> = <font color="Maroon" title="c13">TBn01</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></span>;<br/>
<a NAME="E24:17_1"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
;<br/>
<a NAME="E25:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A21</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c14">LTBn01</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c14">LTBn01</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T52" title="RLTOPSP1:th.52">RLTOPSP1:52</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">yy0</font> = <font color="Maroon" title="c11">y0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></span>;<br/>
<a NAME="E27:17_1"/><span class="lab"><font color="Green" title="E28">A22</font></span>: 
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>) is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></span>;<br/>
<a NAME="E28:17_1"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">y0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c11">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17_1"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="ref" href="normsp_1.html#T6" title="NORMSP_1:th.6">NORMSP_1:6</a></span>;<br/>
<a NAME="E29:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<font color="Maroon" title="c11">y0</font>,<font color="Maroon" title="c10">r</font>)
 
;<br/>
<a NAME="E30:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">y0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c13">TBn01</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:17_1"><span class="lab"><font color="Green" title="E26">A20</font></span></a></span>;<br/>
<a NAME="E31:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">yy0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c14">LTBn01</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T10" target="_self" title="LOPBAN_6:th.10">Th10</a></span>;<br/>
<a NAME="E32:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A23</font></span>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">yy0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c14">LTBn01</font></span>)</span>
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">nb1</font> = 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">TBX1</font> = <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,1)</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">my0</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">yy0</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">TBnx01</font> = <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">BYyr</font> =  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<font color="Maroon" title="c11">y0</font>,<font color="Maroon" title="c10">r</font>) as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">BYr</font> =  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c10">r</font>) as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">XTB01</font> = <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<span class="lab"><font color="Green" title="E30">A24</font></span>:  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">yy0</font> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">yy0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">y0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c11">y0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c14">LTBn01</font> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c22">XTB01</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:17_1"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="ref" href="lopban_6.html#T9" target="_self" title="LOPBAN_6:th.9">Th9</a></span>
<br/>.= 
<font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p2">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T5" target="_self" title="LOPBAN_6:th.5">Th5</a></span>
<br/>.= 
<font color="Maroon" title="c14">LTBn01</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:17_1"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="ref" href="lopban_6.html#T11" target="_self" title="LOPBAN_6:th.11">Th11</a></span>
;<br/>
<a NAME="E42:17_1"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">yy0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c14">LTBn01</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:17_1"><span class="lab"><font color="Green" title="E27">A21</font></span></a>, <a class="txt" href="#E32:17_1"><span class="lab"><font color="Green" title="E29">A23</font></span></a>, <a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>;<br/>
<a NAME="E43:17_1"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">yy0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c13">TBn01</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T10" target="_self" title="LOPBAN_6:th.10">Th10</a></span>;<br/>
<a NAME="E44:17_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">yy0</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c13">TBn01</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E45:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A25</font></span>: 
<font color="Maroon" title="c18">my0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c19">TBnx01</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:17_1"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="ref" href="lopban_6.html#T10" target="_self" title="LOPBAN_6:th.10">Th10</a></span>;<br/>
<a NAME="E46:17_1"/>
<font color="Maroon" title="c20">BYyr</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c19">TBnx01</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:17_1"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="txt" href="#E22:17_1"><span class="lab"><font color="Green" title="E26">A20</font></span></a>, <a class="ref" href="lopban_6.html#T10" target="_self" title="LOPBAN_6:th.10">Th10</a></span>;<br/>
<a NAME="E47:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">my0</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c20">BYyr</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c19">TBnx01</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c19">TBnx01</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:17_1"><span class="lab"><font color="Green" title="E31">A25</font></span></a>, <a class="ref" href="rltopsp1.html#T11" title="RLTOPSP1:th.11">RLTOPSP1:11</a></span>;<br/>
<a NAME="E48:17_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">yy0</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c20">BYyr</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c19">TBnx01</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c19">TBnx01</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_4.html#T33" title="RUSUB_4:th.33">RUSUB_4:33</a></span>;<br/>
<a NAME="E49:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A26</font></span>: 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c11">y0</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<font color="Maroon" title="c11">y0</font>,<font color="Maroon" title="c10">r</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c19">TBnx01</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c19">TBnx01</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:17_1"><span class="lab"><font color="Green" title="E30">A24</font></span></a>, <a class="ref" href="lopban_6.html#T8" target="_self" title="LOPBAN_6:th.8">Th8</a></span>;<br/>
<a NAME="E50:17_1"/><span class="lab"><font color="Green" title="E33">A27</font></span>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c19">TBnx01</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c19">TBnx01</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">TBnx01</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c19">TBnx01</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T62" title="RLTOPSP1:th.62">RLTOPSP1:62</a></span>;<br/>
<a NAME="E51:17_1"/>
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<font color="Maroon" title="c11">y0</font>,<font color="Maroon" title="c10">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y0</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c10">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T2" target="_self" title="LOPBAN_6:th.2">Th2</a></span>;<br/>
<a NAME="E52:17_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">BYyr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">yy0</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c21">BYr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T8" target="_self" title="LOPBAN_6:th.8">Th8</a></span>;<br/>
<a NAME="E53:17_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">yy0</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c20">BYyr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">yy0</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c15">yy0</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c21">BYr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T6" title="RLTOPSP1:th.6">RLTOPSP1:6</a></span>;<br/>
<a NAME="E54:17_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">yy0</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c20">BYyr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c21">BYr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></span>;<br/>
<a NAME="E55:17_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">yy0</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c20">BYyr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="normsp_2.html#K4" title="NORMSP_2:func.4">LinearTopSpaceNorm</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c21">BYr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_4.html#T33" title="RUSUB_4:th.33">RUSUB_4:33</a></span>;<br/>
<a NAME="E56:17_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c15">yy0</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c20">BYyr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">BYr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T35" title="CONVEX1:th.35">CONVEX1:35</a></span>;<br/>
<a NAME="E57:17_1"/><span class="kw">then </span>
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c10">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c11">y0</font></span>)</span> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<font color="Maroon" title="c11">y0</font>,<font color="Maroon" title="c10">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:17_1"><span class="lab"><font color="Green" title="E30">A24</font></span></a>, <a class="ref" href="lopban_6.html#T8" target="_self" title="LOPBAN_6:th.8">Th8</a></span>;<br/>
<a NAME="E58:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A28</font></span>: 
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c10">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">TBnx01</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c19">TBnx01</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:17_1"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="txt" href="#E50:17_1"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E59:17_1"/>
<font color="Maroon" title="c19">TBnx01</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c19">TBnx01</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T32" title="CONVEX1:th.32">CONVEX1:32</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E35">A29</font></span>: <font color="Maroon" title="c19">TBnx01</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c19">TBnx01</font> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c19">TBnx01</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T13" target="_self" title="LOPBAN_6:th.13">Th13</a>, <a class="ref" href="convex1.html#T13" title="CONVEX1:th.13">CONVEX1:13</a></span>
<br/>.= 
2 <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c19">TBnx01</font>

;<br/>
<a NAME="E61:17_1"/>
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T3" target="_self" title="LOPBAN_6:th.3">Th3</a></span>;<br/>
<a NAME="E62:17_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c17">TBX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T5" target="_self" title="LOPBAN_6:th.5">Th5</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c19">TBnx01</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c19">TBnx01</font> = 
2 <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c5">TB1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:17_1"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="ref" href="lopban_6.html#T9" target="_self" title="LOPBAN_6:th.9">Th9</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c5">TB1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T37" title="CONVEX1:th.37">CONVEX1:37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c5">TB1</font>

;<br/>
<a NAME="E64:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A30</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">TBnx01</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c19">TBnx01</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">TB1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T52" title="RLTOPSP1:th.52">RLTOPSP1:52</a></span>;<br/>
<a NAME="E65:17_1"/><span class="lab"><font color="Green" title="E37">A31</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17_1"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/>
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>) = 
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>)

<br/>.= 
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T74" title="XCMPLX_1:th.74">XCMPLX_1:74</a></span>
<br/>.= 
<font color="Maroon" title="c16">nb1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c10">r</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T3" target="_self" title="LOPBAN_6:th.3">Th3</a></span>
<br/>.= 
<font color="Maroon" title="c16">nb1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c21">BYr</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T9" target="_self" title="LOPBAN_6:th.9">Th9</a></span>
;<br/>
<a NAME="E67:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A32</font></span>: 
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">TB1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:17_1"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E64:17_1"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="ref" href="convex1.html#T39" title="CONVEX1:th.39">CONVEX1:39</a></span>;<br/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">TB1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">TB1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T37" title="CONVEX1:th.37">CONVEX1:37</a></span>
<br/>.= 
1 <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">TB1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T106" title="XCMPLX_1:th.106">XCMPLX_1:106</a></span>
<br/>.= 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">TB1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T32" title="CONVEX1:th.32">CONVEX1:32</a></span>
;<br/>
<a NAME="E69:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A33</font></span>: 
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17_1"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="txt" href="#E67:17_1"><span class="lab"><font color="Green" title="E38">A32</font></span></a>, <a class="txt" href="#E27:17_1"><span class="lab"><font color="Green" title="E28">A22</font></span></a>, <a class="ref" href="lopban_6.html#T15" target="_self" title="LOPBAN_6:th.15">Th15</a>, <a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/>
<a NAME="E70:17_1"/><span class="lab"><font color="Green" title="E40">A34</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">q</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">q</font> &amp;  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Olive" title="b2">q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Olive" title="b1">p</font>)</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c23">TB1</font> = <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,1)</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c24">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">q</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> &amp;  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Olive" title="b1">q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c24">p</font>)</span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:17_1_10"/><span class="lab"><font color="Green" title="E41">A35</font></span>: 
<font color="Maroon" title="c24">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> &amp;  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Olive" title="b1">q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c24">p</font>)</span>)</span> )</span><br/>

<a NAME="E3:17_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A36</font></span>: 
<font color="Maroon" title="c24">p</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">p</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T3" target="_self" title="LOPBAN_6:th.3">Th3</a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c24">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c24">p</font> &amp;  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c24">p</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c24">p</font>)</span>)</span> )</span><br/>

<a NAME="E4:17_1_10"/>
<font color="Maroon" title="c24">p</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">p</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c23">TB1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:17_1"><span class="lab"><font color="Green" title="E39">A33</font></span></a>, <a class="ref" href="convex1.html#T39" title="CONVEX1:th.39">CONVEX1:39</a></span>;<br/>
<a NAME="E5:17_1_10"/><span class="kw">then </span>
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c24">p</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">p</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c23">TB1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_10"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="ref" href="lopban_6.html#T3" target="_self" title="LOPBAN_6:th.3">Th3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:17_1_10"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c24">p</font> &amp;  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n0</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c24">p</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c24">p</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E65:17_1"><span class="lab"><font color="Green" title="E37">A31</font></span></a>, <a class="txt" href="#E2:17_1_10"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="txt" href="#E3:17_1_10"><span class="lab"><font color="Green" title="E42">A36</font></span></a>, <a class="ref" href="lopban_6.html#T5" target="_self" title="LOPBAN_6:th.5">Th5</a>, <a class="ref" href="xreal_1.html#T129" title="XREAL_1:th.129">XREAL_1:129</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E71:17_1"/><span class="lab"><font color="Green" title="E41">A37</font></span>: 
<font color="Maroon" title="c6">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_11"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font> implies  ex <font color="Olive" title="b1">q</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">w</font> where <font color="Olive" title="b2">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font> : <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b2">w</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:17_1_11"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">w</font> where <font color="Olive" title="b2">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font> : <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b2">w</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E3:17_1_11"/><span class="lab"><font color="Green" title="E42">A38</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">and </span><br/><a NAME="E4:17_1_11"/><span class="lab"><font color="Green" title="E43">A39</font></span>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="funct_2.html#T65" title="FUNCT_2:th.65">FUNCT_2:65</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c25">p</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:17_1_11"/><span class="lab"><font color="Green" title="E44">A40</font></span>: 
<font color="Maroon" title="c25">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E7:17_1_11"/><span class="lab"><font color="Green" title="E45">A41</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:17_1"><span class="lab"><font color="Green" title="E41">A37</font></span></a>, <a class="txt" href="#E3:17_1_11"><span class="lab"><font color="Green" title="E42">A38</font></span></a>, <a class="ref" href="normsp_2.html#T22" title="NORMSP_2:th.22">NORMSP_2:22</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">TBp</font> = <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c25">p</font>)</span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c27">q</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E10:17_1_11"/><span class="lab"><font color="Green" title="E46">A42</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q</font>
 <span class="kw">and </span><br/><a NAME="E11:17_1_11"/><span class="lab"><font color="Green" title="E47">A43</font></span>: 
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c27">q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c26">TBp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:17_1"><span class="lab"><font color="Green" title="E40">A34</font></span></a>, <a class="txt" href="#E6:17_1_11"><span class="lab"><font color="Green" title="E44">A40</font></span></a></span>;<br/><a NAME="E12:17_1_11"/>
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">p</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1_11"><span class="lab"><font color="Green" title="E45">A41</font></span></a></span>;<br/><a NAME="E13:17_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A44</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c25">p</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T2" target="_self" title="LOPBAN_6:th.2">Th2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_11_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c28">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c26">TBp</font> implies <font color="Maroon" title="c28">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_1_11_1"/>
<font color="Maroon" title="c28">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c26">TBp</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c28">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">tz0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E3:17_1_11_1"/><span class="lab"><font color="Green" title="E49">A45</font></span>: 
<font color="Maroon" title="c28">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c29">tz0</font>
 <span class="kw">and </span><br/><a NAME="E4:17_1_11_1"/><span class="lab"><font color="Green" title="E50">A46</font></span>: 
<font color="Maroon" title="c29">tz0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">TBp</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c30">z0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E6:17_1_11_1"/><span class="lab"><font color="Green" title="E51">A47</font></span>: 
<font color="Maroon" title="c30">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c25">p</font>)
 <span class="kw">and </span><br/><a NAME="E7:17_1_11_1"/><span class="lab"><font color="Green" title="E52">A48</font></span>: 
<font color="Maroon" title="c29">tz0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">z0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_11_1"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="ref" href="funct_2.html#T65" title="FUNCT_2:th.65">FUNCT_2:65</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">z0</font> = <font color="Maroon" title="c30">z0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E9:17_1_11_1"/><span class="lab"><font color="Green" title="E53">A49</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c31">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">x</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c25">p</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_11_1"><span class="lab"><font color="Green" title="E51">A47</font></span></a></span>;<br/><a NAME="E10:17_1_11_1"/>
<font color="Maroon" title="c28">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c31">z0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_11"><span class="lab"><font color="Green" title="E43">A39</font></span></a>, <a class="txt" href="#E3:17_1_11_1"><span class="lab"><font color="Green" title="E49">A45</font></span></a>, <a class="txt" href="#E7:17_1_11_1"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="ref" href="grcat_1.html#D8" title="GRCAT_1:def.8">GRCAT_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E11:17_1_11_1"/>
<font color="Maroon" title="c28">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E13:17_1_11"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="txt" href="#E9:17_1_11_1"><span class="lab"><font color="Green" title="E53">A49</font></span></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E15:17_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A50</font></span>: 
<font color="Maroon" title="c23">y</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c26">TBp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c28">q</font> = <font color="Maroon" title="c27">q</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">q</font> &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font> : <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">w</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">q</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_11_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c29">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c28">q</font>)</span>)</span> implies <font color="Maroon" title="c29">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c26">TBp</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_1_11_2"/>
<font color="Maroon" title="c29">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c28">q</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c26">TBp</font></span><br/><a NAME="E2:17_1_11_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">z0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c29">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b1">z0</font> &amp; <font color="Olive" title="b1">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c28">q</font>) )
 ;<br/><span class="kw">hence </span><a NAME="E3:17_1_11_2"/>
<font color="Maroon" title="c29">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">y</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c26">TBp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:17_1_11"><span class="lab"><font color="Green" title="E47">A43</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E17:17_1_11"/><span class="kw">then </span>
<font color="Maroon" title="c23">y</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">Y</font></span>)</span>,<font color="Maroon" title="c28">q</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">y</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c26">TBp</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E18:17_1_11"/><span class="kw">then </span>
 <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> (<font color="Maroon" title="c23">y</font>,<font color="Maroon" title="c28">q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">y</font> <a href="rusub_4.html#K5" title="RUSUB_4:func.5">+</a> <font color="Maroon" title="c26">TBp</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_6.html#T2" target="_self" title="LOPBAN_6:th.2">Th2</a></span>;<br/><span class="kw">hence </span><a NAME="E19:17_1_11"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">q</font> &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font> : <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">w</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">q</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E15:17_1_11"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="txt" href="#E10:17_1_11"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E73:17_1"/>
<font color="Maroon" title="c4">T1</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_2.html#T22" title="NORMSP_2:th.22">NORMSP_2:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
