<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K5" target="_self">RealVectSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>





<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K2" target="_self">FuncAdd</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="lopban_1.html#E1:23_1"><i><font color="Green">E19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K2" target="_self">FuncAdd</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>

;<br/><b>hence </b><a NAME="E3:23_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lopban_1.html#K6" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_1.html#K6" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_1.html#K6" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T3" target="_self">Th3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:23"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>5</sub></font> <a href="lopban_1.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lopban_1.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lopban_1.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E4:23"/><b>then </b><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
;<br/>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="lopban_1.html#K2" target="_self">FuncAdd</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="lopban_1.html#K2" target="_self">FuncAdd</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funcsdom.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>

;<br/>
<b>hence </b><a NAME="E6:23"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="lopban_1.html#E4:23"><i><font color="Green">E19</font></i></a>, <a class="ref" href="lopban_1.html#T3" target="_self">Th3</a></i>;<br/>


</div>
