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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">M1</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>  &amp; <font color="Olive" title="b2">M2</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>  &amp; <font color="Olive" title="b3">M3</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</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="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> </span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c2">M1</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>  &amp; <font color="Maroon" title="c3">M2</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>  &amp; <font color="Maroon" title="c4">M3</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</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="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">r1</font>, <font color="Maroon" title="c6">r2</font>, <font color="Maroon" title="c7">r3</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">M1</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>  &amp; <font color="Maroon" title="c3">M2</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>  &amp; <font color="Maroon" title="c4">M3</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</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="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
( <font color="Maroon" title="c2">M1</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>  &amp; <font color="Maroon" title="c3">M2</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>  )
 <span class="kw">and </span><br/><a NAME="E2:7"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c4">M3</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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"><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="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> </span><br/>

<a NAME="E3:7"/>
<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="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="circled1.html#T3" target="_self" onmouseover="rs('circled1/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E4:7"/><span class="kw">then </span>
<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="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="circled1.html#T3" target="_self" onmouseover="rs('circled1/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:7"/>
<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="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T32" onmouseover="rs('convex1/T32')" onmouseout="rh()">CONVEX1:32</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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