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<span class="kw">let </span><font color="Maroon" title="c1">r</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">  for <font color="Olive" title="b1">R</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>   <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>   <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>   <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>   <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">RLSStruct</a> <br/>  for <font color="Olive" title="b2">AR</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">R</font>  st <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b2">AR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Olive" title="b2">AR</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>   <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>   <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>   <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>   <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">RLSStruct</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">AR</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</font>  st <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b1">AR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Olive" title="b1">AR</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">AR</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">R</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="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">AR</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:78"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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">  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">AR</font></span>)</span></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T10" target="_self" title="RLAFFIN1:th.10">Th10</a></span>
<br/>.= 
1 <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c3">AR</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T11" target="_self" title="RLAFFIN1:th.11">Th11</a></span>
;<br/>
<a NAME="E3:78"/><span class="kw">then </span>
<font color="Maroon" title="c3">AR</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin1.html#E61"><span class="lab"><font color="Green" title="E50">Lm7</font></span></a>, <a class="ref" href="rlaffin1.html#T9" target="_self" title="RLAFFIN1:th.9">Th9</a></span>;<br/>
<a NAME="E4:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A2</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">AR</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T51" target="_self" title="RLAFFIN1:th.51">Th51</a>, <a class="ref" href="rlaffin1.html#T54" target="_self" title="RLAFFIN1:th.54">Th54</a></span>;<br/>
<a NAME="E5:78"/>
<font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">AR</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin1.html#E61"><span class="lab"><font color="Green" title="E50">Lm7</font></span></a>, <a class="ref" href="rlaffin1.html#T9" target="_self" title="RLAFFIN1:th.9">Th9</a></span>;<br/>
<a NAME="E6:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A3</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">AR</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T51" target="_self" title="RLAFFIN1:th.51">Th51</a>, <a class="ref" href="rlaffin1.html#T54" target="_self" title="RLAFFIN1:th.54">Th54</a></span>;<br/>
<font color="Maroon" title="c1">r</font> <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="c1">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p2">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a></span>)</span></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T10" target="_self" title="RLAFFIN1:th.10">Th10</a></span>
<br/>.= 
1 <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
<br/>.= 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T11" target="_self" title="RLAFFIN1:th.11">Th11</a></span>
;<br/>
<a NAME="E8:78"/><span class="kw">then </span>
<font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">AR</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="rlaffin1.html#T9" target="_self" title="RLAFFIN1:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:78"/>
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c3">AR</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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/>


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