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<b>let </b><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <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">e1</font>, <font color="Olive" title="b2">e2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c2">o1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b1">e1</font></span>)</span> <a href="convex4.html#K5" title="CONVEX4:func.5">+</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b2">e2</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b1">e1</font></span>)</span> <a href="convex4.html#K5" title="CONVEX4:func.5">+</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b2">e2</font></span>)</span> ) implies <font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font> )</span><br/>

<b>assume </b><a NAME="E1:55_1_2"/><i><font color="Green" title="E35">A2</font></i>: 
for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c2">o1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b1">e1</font></span>)</span> <a href="convex4.html#K5" title="CONVEX4:func.5">+</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b2">e2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font> st not <font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b1">e1</font></span>)</span> <a href="convex4.html#K5" title="CONVEX4:func.5">+</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b2">e2</font></span>)</span> or <font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font> )</span><br/>

<b>assume </b><a NAME="E2:55_1_2"/><i><font color="Green" title="E36">A3</font></i>: 
for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b1">e1</font></span>)</span> <a href="convex4.html#K5" title="CONVEX4:func.5">+</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b2">e2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">e1</font>, <font color="Maroon" title="c5">e2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <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"> <font color="Maroon" title="c2">o1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font></span><br/><b>thus </b><font color="Maroon" title="c2">o1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font> = 
<span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Maroon" title="c4">e1</font></span>)</span> <a href="convex4.html#K5" title="CONVEX4:func.5">+</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Maroon" title="c5">e2</font></span>)</span>
<b>by </b><i><a class="txt" href="convex4.html#E1:55_1_2"><i><font color="Green" title="E35">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">o2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">e1</font>,<font color="Maroon" title="c5">e2</font>
<b>by </b><i><a class="txt" href="convex4.html#E2:55_1_2"><i><font color="Green" title="E36">A3</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:55_1_2"/>
<font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</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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