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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/> for <font color="Olive" title="b3">T</font> being  <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="b4">V</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="c2">Y</font></span>)</span>  st <font color="Olive" title="b4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">T</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font></span>)</span> holds <br/><font color="Olive" title="b4">V</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a></span><br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/> for <font color="Olive" title="b2">T</font> being  <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="b3">V</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="c2">Y</font></span>)</span>  st <font color="Olive" title="b3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">T</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">r</font></span>)</span> holds <br/><font color="Olive" title="b3">V</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a></span><br/><b>let </b><font color="Maroon" title="c4">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being  <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">V</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="c2">Y</font></span>)</span>  st <font color="Olive" title="b2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font></span>)</span> holds <br/><font color="Olive" title="b2">V</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a></span><br/><b>let </b><font color="Maroon" title="c5">T</font> be   <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</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="c2">Y</font></span>)</span>  st <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font></span>)</span> holds <br/><font color="Olive" title="b1">V</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a></span><br/><b>let </b><font color="Maroon" title="c6">V</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="c2">Y</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font></span>)</span> implies <font color="Maroon" title="c6">V</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> )</span><br/>











<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c6">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">V</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">V1</font> = <font color="Maroon" title="c5">T</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</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/>
<a NAME="E3:14"/><i><font color="Green" title="E13">K2</font></i>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font><br/> for <font color="Olive" title="b3">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <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="b3">s</font> &amp; <font color="Olive" title="b3">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">u</font>, <font color="Maroon" title="c9">v</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <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">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font></span><br/><b>let </b><font color="Maroon" title="c10">s</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c10">s</font> &amp; <font color="Maroon" title="c10">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font> )</span><br/>



<b>assume </b><a NAME="E1:14_1"/><i><font color="Green" title="E14">A2</font></i>: 
(  <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">s</font> &amp; <font color="Maroon" title="c10">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font></span><br/>

<b>consider </b><font color="Maroon" title="c11">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_1"/><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c11">z1</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="c3">x</font>,<font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z1</font> )
 <b>by </b><i><a class="txt" href="#E1:14_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">zz1</font> = <font color="Maroon" title="c11">z1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="#E3:14_1"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/>
<a NAME="E5:14_1"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">zz1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:14_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="lopban_1.html#D6" title="LOPBAN_1:def.6">LOPBAN_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:14_1"/><i><font color="Green" title="E17">A7</font></i>: 
( <font color="Maroon" title="c13">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c13">z2</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="c3">x</font>,<font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z2</font> )
 <b>by </b><i><a class="txt" href="#E1:14_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">zz2</font> = <font color="Maroon" title="c13">z2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E17">A7</font></i></a></i>;<br/>
<a NAME="E9:14_1"/><i><font color="Green" title="E18">A10</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">zz2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:14_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="lopban_1.html#D6" title="LOPBAN_1:def.6">LOPBAN_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">za1</font> = <font color="Maroon" title="c12">zz1</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="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">za2</font> = <font color="Maroon" title="c14">zz2</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="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">Bxr</font> =  <a href="lopban_5.html#K1" title="LOPBAN_5:func.1">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<a NAME="E13:14_1"/>
<font color="Maroon" title="c17">Bxr</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 
<b>by </b><i><a class="ref" href="lopban_6.html#T12" target="_self" title="LOPBAN_6:th.12">Lm11</a></i>;<br/>
<a NAME="E14:14_1"/><b>then </b><i><font color="Green" title="E19">AM1</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">za1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">za2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Bxr</font>
 
<b>by </b><i><a class="txt" href="#E1:14_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E3:14_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E7:14_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="convex1.html#D2" title="CONVEX1:def.2">CONVEX1:def 2</a></i>;<br/>
<a NAME="E15:14_1"/><i><font color="Green" title="E20">AM2</font></i>: 
<font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">za1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">zz1</font>
 
<b>by </b><i><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<a NAME="E16:14_1"/><i><font color="Green" title="E21">AM3</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">za2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">zz2</font>
 
<b>by </b><i><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<a NAME="E17:14_1"/><i><font color="Green" title="E22">AM4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">za1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">za2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">zz1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">zz2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E15:14_1"><i><font color="Green" title="E20">AM2</font></i></a>, <a class="txt" href="#E16:14_1"><i><font color="Green" title="E21">AM3</font></i></a>, <a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<a NAME="E18:14_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">zz1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">zz2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:14_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="#E9:14_1"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="lopban_1.html#D5" title="LOPBAN_1:def.5">LOPBAN_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E19:14_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V1</font>
 <b>by </b><i><a class="txt" href="#E14:14_1"><i><font color="Green" title="E19">AM1</font></i></a>, <a class="txt" href="#E17:14_1"><i><font color="Green" title="E22">AM4</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:14"/>
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <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><br/> for <font color="Olive" title="b3">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <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="b3">s</font> &amp; <font color="Olive" title="b3">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">u</font>, <font color="Maroon" title="c9">v</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <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">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font></span><br/><b>let </b><font color="Maroon" title="c10">s</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c10">s</font> &amp; <font color="Maroon" title="c10">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> )</span><br/>



<b>assume </b><a NAME="E1:14_2"/><i><font color="Green" title="E14">ASV</font></i>: 
(  <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">s</font> &amp; <font color="Maroon" title="c10">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">u1</font> = <font color="Maroon" title="c8">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">v1</font> = <font color="Maroon" title="c9">v</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<a NAME="E4:14_2"/><i><font color="Green" title="E15">L2</font></i>: 
<font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font>
 
<b>by </b><i><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<a NAME="E5:14_2"/><i><font color="Green" title="E16">L3</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font>
 
<b>by </b><i><a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<a NAME="E6:14_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">v1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:14_2"><i><font color="Green" title="E15">L2</font></i></a>, <a class="txt" href="#E5:14_2"><i><font color="Green" title="E16">L3</font></i></a>, <a class="ref" href="normsp_2.html#D4" title="NORMSP_2:def.4">NORMSP_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:14_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font>
 <b>by </b><i><a class="txt" href="#E3:14"><i><font color="Green" title="E13">K2</font></i></a>, <a class="txt" href="#E1:14"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E1:14_2"><i><font color="Green" title="E14">ASV</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:14"/>
<font color="Maroon" title="c6">V</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 <b>by </b><i><a class="ref" href="convex1.html#D2" title="CONVEX1:def.2">CONVEX1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
