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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="Olive" title="b2">AR</font> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a>  holds <br/><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b2">AR</font> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> </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="Olive" title="b1">AR</font> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a>  holds <br/><font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b1">AR</font> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> </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="c3">AR</font> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a>  implies <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:77"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c3">AR</font> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</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="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font> is  <a href="rusub_4.html#V2" title="RUSUB_4:attr.2">Affine</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">R</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rusub_4.html#D4" title="RUSUB_4:def.4">RUSUB_4:def 4</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">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   holds <br/>(  not <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font> or  not <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font> or <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">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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><br/><span class="kw">let </span><font color="Maroon" title="c6">s</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"> (  not <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font> or  not <font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font> or <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="c6">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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><br/>



<span class="kw">assume </span><a NAME="E2:77"/>
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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 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="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">AR</font> or <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="c6">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">w1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font><span class="kw"> such that </span><br/><a NAME="E4:77"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w1</font>
 <span class="kw">and </span><br/><a NAME="E5:77"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
<font color="Maroon" title="c7">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AR</font>
 ;<br/>
<span class="kw">assume </span><a NAME="E6:77"/>
<font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c6">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">w2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R</font><span class="kw"> such that </span><br/><a NAME="E8:77"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<font color="Maroon" title="c5">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w2</font>
 <span class="kw">and </span><br/><a NAME="E9:77"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
<font color="Maroon" title="c8">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AR</font>
 ;<br/>
<a NAME="E10:77"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
<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="c6">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">AR</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E5:77"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E9:77"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="ref" href="rusub_4.html#D4" title="RUSUB_4:def.4">RUSUB_4:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E62">A7</font></span>: <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w1</font></span>)</span> = 
<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="c6">s</font></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="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</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="c6">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
;<br/>
<font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
;<br/>
<a NAME="E13:77"/><span class="kw">then </span>
<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="c6">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> <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="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w1</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="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E8:77"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="txt" href="#E11:77"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:77"/>
<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="c6">s</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v1</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E10:77"><span class="lab"><font color="Green" title="E61">A6</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>
