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<span class="kw">let </span><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 <span class="p1">(<span class="default"><a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:55_1_2"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E2:55_1_2"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_2"><span class="lab"><font color="Green" title="E35">A2</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55_1_2"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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