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<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">RLSStruct</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">M1</font>, <font color="Olive" title="b2">M2</font>, <font color="Olive" title="b3">M3</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b4">r1</font>, <font color="Olive" title="b5">r2</font>, <font color="Olive" title="b6">r3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">M1</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Olive" title="b2">M2</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Olive" title="b3">M3</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b1">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b2">M2</font></span>)</span></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">r3</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b3">M3</font></span>)</span> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a></span><br/><b>let </b><font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c3">M2</font>, <font color="Maroon" title="c4">M3</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</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">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r3</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c2">M1</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Maroon" title="c3">M2</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Maroon" title="c4">M3</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">r3</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">M3</font></span>)</span> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a></span><br/><b>let </b><font color="Maroon" title="c5">r1</font>, <font color="Maroon" title="c6">r2</font>, <font color="Maroon" title="c7">r3</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"> ( <font color="Maroon" title="c2">M1</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Maroon" title="c3">M2</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> &amp; <font color="Maroon" title="c4">M3</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r3</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">M3</font></span>)</span> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">M1</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c3">M2</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c4">M3</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 ; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r3</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">M3</font></span>)</span> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a></span><br/>

<a NAME="E4:26"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 
<b>by </b><i><a class="txt" href="convex1.html#E1:26"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="convex1.html#E2:26"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="convex1.html#T11" target="_self" title="CONVEX1:th.11">Th11</a></i>;<br/>
<a NAME="E5:26"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span></span>)</span></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r3</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">M3</font></span>)</span> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 
<b>by </b><i><a class="txt" href="convex1.html#E3:26"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="convex1.html#T11" target="_self" title="CONVEX1:th.11">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" title="c5">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r3</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">M3</font></span>)</span> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 <b>by </b><i><a class="txt" href="convex1.html#E9"><i><font color="Green" title="E3">Lm1</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/>


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