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<span class="kw">let </span><font color="Maroon" title="c4">V</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">RLSStruct</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   <a href="convex3.html#V1" title="CONVEX3:attr.1">cone</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">V</font>  st <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp; <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp; <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp; <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  holds <br/>( <font color="Olive" title="b1">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  iff  for <font color="Olive" title="b2">u</font>, <font color="Olive" title="b3">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>  st <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">M</font> holds <br/><font color="Olive" title="b2">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">M</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">M</font> be   <a href="convex3.html#V1" title="CONVEX3:attr.1">cone</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp; <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp; <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp; <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  implies ( <font color="Maroon" title="c5">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  iff  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>  st <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> ) )</span><br/>



<a NAME="E1:14"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
( (  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>  st <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> ) implies <font color="Maroon" title="c5">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>  st <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> </span><br/>

<a NAME="E2:14_1"/>
 for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font><br/>  for <font color="Olive" title="b3">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b3">r</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">r</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">u</font>, <font color="Maroon" title="c7">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c6">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Maroon" title="c7">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font> &amp; <font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c6">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Maroon" title="c7">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14_1_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">r</font>
 <span class="kw">and </span><br/><a NAME="E2:14_1_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c8">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">and </span><br/><a NAME="E3:14_1_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 <span class="kw">and </span><br/><a NAME="E4:14_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c8">r</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E6:14_1_1"/>
<font color="Maroon" title="c9">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a></span>;<br/>
<a NAME="E7:14_1_1"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/>
<a NAME="E8:14_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="convex3.html#D3" target="_self" onmouseover="rs('convex3/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E9:14_1_1"/>
<font color="Maroon" title="c9">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E3:14_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="convex3.html#D3" target="_self" onmouseover="rs('convex3/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:14_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:14_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:14_1"/>
<font color="Maroon" title="c5">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#D2" onmouseover="rs('convex1/D2')" onmouseout="rh()">CONVEX1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:14"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
( <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp; <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp; <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp; <font color="Maroon" title="c4">V</font> is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  iff  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>  st <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> )</span><br/>

<a NAME="E3:14"/>
( <font color="Maroon" title="c5">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  implies  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>  st <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:14_2"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c5">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>  st <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font></span><br/>

<a NAME="E2:14_2"/>
 for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>  st <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">u</font>, <font color="Maroon" title="c7">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Maroon" title="c7">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> implies <font color="Maroon" title="c6">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_2_1"/>
( <font color="Maroon" title="c6">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Maroon" title="c7">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font></span><br/>

<a NAME="E2:14_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_2"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="convex1.html#D2" onmouseover="rs('convex1/D2')" onmouseout="rh()">CONVEX1:def 2</a></span>;<br/>
<a NAME="E3:14_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A10</font></span>: 
2 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">jd</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">jd</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex3.html#D3" target="_self" onmouseover="rs('convex3/D3')" onmouseout="rh()">Def3</a></span>;<br/>
2 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default">2 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default">2 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="rlvect_1.html#D5" onmouseover="rs('rlvect_1/D5')" onmouseout="rh()">RLVECT_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default">2 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="rlvect_1.html#D7" onmouseover="rs('rlvect_1/D7')" onmouseout="rh()">RLVECT_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="rlvect_1.html#D7" onmouseover="rs('rlvect_1/D7')" onmouseout="rh()">RLVECT_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c6">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default">1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="rlvect_1.html#D8" onmouseover="rs('rlvect_1/D8')" onmouseout="rh()">RLVECT_1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:14_2_1"/>
<font color="Maroon" title="c6">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E3:14_2_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="ref" href="rlvect_1.html#D8" onmouseover="rs('rlvect_1/D8')" onmouseout="rh()">RLVECT_1:def 8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:14_2"/>
 for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>  st <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:14"/>
( <font color="Maroon" title="c5">M</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  iff  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c4">V</font>  st <font color="Olive" title="b1">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> &amp; <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> holds <br/><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
