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

<span class="kw">let </span><font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c2">S</font> be    <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"> ( ( <font color="Maroon" title="c2">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  ) &amp; ( <font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  implies <font color="Maroon" title="c2">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> ) &amp; ( <font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  ) &amp; ( <font color="Maroon" title="c2">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  implies <font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:7"/>
( <font color="Maroon" title="c2">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  )
  <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">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  implies <font color="Maroon" title="c2">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> ) &amp; ( <font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  ) &amp; ( <font color="Maroon" title="c2">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  implies <font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7_1"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E2:7_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a> <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">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:7_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:7_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="waybel10.html#T1" target="_self" title="WAYBEL10:th.1">Th1</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a> <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">thus </span><a NAME="E2:7"/>
( <font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  implies <font color="Maroon" title="c2">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</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">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> iff <font color="Maroon" title="c2">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7_2"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:7_2"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a> <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">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">thus </span><a NAME="E3:7_2"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_2"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="txt" href="#E2:7_2"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="ref" href="waybel10.html#T1" target="_self" title="WAYBEL10:th.1">Th1</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a> <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">thus </span><a NAME="E3:7"/>
( <font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  )
  <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">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  implies <font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7_3"/><span class="lab"><font color="Green" title="E3">A5</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E2:7_3"/><span class="lab"><font color="Green" title="E4">A6</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a> <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">S</font> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span><br/>

<span class="kw">thus </span><a NAME="E3:7_3"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_3"><span class="lab"><font color="Green" title="E3">A5</font></span></a>, <a class="txt" href="#E2:7_3"><span class="lab"><font color="Green" title="E4">A6</font></span></a>, <a class="ref" href="waybel10.html#T1" target="_self" title="WAYBEL10:th.1">Th1</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a> <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">assume </span><span class="kw">that </span><br/><a NAME="E4:7"/><span class="lab"><font color="Green" title="E3">A7</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:7"/><span class="lab"><font color="Green" title="E4">A8</font></span>: 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a> <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">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">thus </span><a NAME="E6:7"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E3">A7</font></span></a>, <a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E4">A8</font></span></a>, <a class="ref" href="waybel10.html#T1" target="_self" title="WAYBEL10:th.1">Th1</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a> <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>
