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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="clvect_1.html#V2" title="CLVECT_1:attr.2">ComplexLinearSpace-like</a>  <a href="clvect_1.html#L1" title="CLVECT_1:struct.1">CLSStruct</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">z1</font>, <font color="Olive" title="b5">z2</font>, <font color="Olive" title="b6">z3</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Olive" title="b1">M1</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> &amp; <font color="Olive" title="b2">M2</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> &amp; <font color="Olive" title="b3">M3</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">z1</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z2</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z3</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Olive" title="b3">M3</font></span>)</span> is <a href="convex4.html#V2" title="CONVEX4:attr.2">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">z1</font>, <font color="Olive" title="b2">z2</font>, <font color="Olive" title="b3">z3</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st <font color="Maroon" title="c2">M1</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> &amp; <font color="Maroon" title="c3">M2</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> &amp; <font color="Maroon" title="c4">M3</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z2</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z3</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c4">M3</font></span>)</span> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a></span><br/><b>let </b><font color="Maroon" title="c5">z1</font>, <font color="Maroon" title="c6">z2</font>, <font color="Maroon" title="c7">z3</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</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="convex4.html#V2" title="CONVEX4:attr.2">convex</a> &amp; <font color="Maroon" title="c3">M2</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> &amp; <font color="Maroon" title="c4">M3</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z2</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z3</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c4">M3</font></span>)</span> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> )</span><br/>





<b>assume </b><a NAME="E1:99"/><i><font color="Green" title="E59">A1</font></i>: 
( <font color="Maroon" title="c2">M1</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> &amp; <font color="Maroon" title="c3">M2</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> &amp; <font color="Maroon" title="c4">M3</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">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">z1</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z2</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z3</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c4">M3</font></span>)</span> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a></span><br/>

<a NAME="E2:99"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z2</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a>
 
<b>by </b><i><a class="txt" href="convex4.html#E1:99"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="convex4.html#T60" target="_self" title="CONVEX4:th.60">Th211</a></i>;<br/>
<a NAME="E3:99"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z2</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z3</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c4">M3</font></span>)</span> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a>
 
<b>by </b><i><a class="txt" href="convex4.html#E1:99"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="convex4.html#T60" target="_self" title="CONVEX4:th.60">Th211</a></i>;<br/>
<b>hence </b><a NAME="E4:99"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">z1</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z2</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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">z3</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c4">M3</font></span>)</span> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a>
 <b>by </b><i><a class="ref" href="convex4.html#T51" target="_self" title="CONVEX4:th.51">Lm201</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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