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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <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">S</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">f</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  holds <br/> <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">f</font> is  <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  st  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">f</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  holds <br/> <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">f</font> is  <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  implies  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:47"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font> is  <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">N</font> =  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_6.html#D5" title="YELLOW_6:def.5">YELLOW_6:def 5</a> <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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font></span>)</span> st <br/>( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<a NAME="E2:47"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#D10" target="_self" title="WAYBEL11:def.10">Def10</a></span>;<br/>
<a NAME="E3:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A2</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font></span>)</span> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T19" target="_self" title="WAYBEL11:th.19">Th19</a></span>;<br/>
<a NAME="E4:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 
;<br/>
<a NAME="E5:47"/>
<span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E7:47"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E8:47"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">and </span><br/><a NAME="E9:47"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:47"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E12:47"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c8">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:47"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="waybel11.html#D10" target="_self" title="WAYBEL11:def.10">Def10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">z</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="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">z</font> &amp; <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">z</font> )</span><br/>

<a NAME="E14:47"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:47"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="waybel11.html#D10" target="_self" title="WAYBEL11:def.10">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:47"/>
( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">z</font> &amp; <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E9:47"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="waybel11.html#D10" target="_self" title="WAYBEL11:def.10">Def10</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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