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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V7">RealLinearSpace-like</a>  <a href="rlvect_1.html#L2">RLSStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="convex1.html#V1" target="_self">convex</a>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="convex1.html#V1" target="_self">convex</a>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="convex1.html#V1" target="_self">convex</a>
 ;<br/>

<a NAME="E4:26"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="convex1.html#V1" target="_self">convex</a>
 
<b>by </b><i><a class="txt" href="convex1.html#E1:26"><i><font color="Green">E16</font></i></a>, <a class="txt" href="convex1.html#E2:26"><i><font color="Green">E17</font></i></a>, <a class="ref" href="convex1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:26"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="convex1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="convex1.html#V1" target="_self">convex</a>
 
<b>by </b><i><a class="txt" href="convex1.html#E3:26"><i><font color="Green">E18</font></i></a>, <a class="ref" href="convex1.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E6:26"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="convex1.html#V1" target="_self">convex</a>
 <b>by </b><i><a class="txt" href="convex1.html#E9"><i><font color="Green">Lemma3</font></i></a></i>;<br/>


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