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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:41_1_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:41_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:41_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E2:41_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #);<br/>
<b>take </b>
 <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #)
;<br/>

<b>thus </b><a NAME="E5:41_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #) iff ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="waybel_9.html#E2:41_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:41_1_1"/>
( the <a href="orders_2.html#U1">InternalRel</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #) <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="toler_1.html#K1">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #) &amp; the <a href="waybel_0.html#U1">mapping</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #) <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> #) )
 ;<br/>


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