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<span class="kw">let </span><font color="Maroon" title="c1">R</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"> for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">S</font></span>)</span><br/> for <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">S</font></span>)</span><br/> for <font color="Olive" title="b4">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> &amp; <font color="Olive" title="b3">x</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b4">y</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Olive" title="b2">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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"> for <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">S</font></span>)</span><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">S</font></span>)</span><br/> for <font color="Olive" title="b3">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> &amp; <font color="Olive" title="b2">x</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Olive" title="b3">y</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Olive" title="b1">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">S</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">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">S</font></span>)</span><br/> for <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Maroon" title="c3">B</font> holds <br/><font color="Olive" title="b2">y</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Maroon" title="c3">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">S</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">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c4">x</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Maroon" title="c3">B</font> holds <br/><font color="Olive" title="b1">y</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Maroon" title="c3">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c4">x</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Maroon" title="c3">B</font> implies <font color="Maroon" title="c5">y</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Maroon" title="c3">B</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:68"/><span class="lab"><font color="Green" title="E38">A</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">and </span><br/><a NAME="E2:68"/><span class="lab"><font color="Green" title="E39">B</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Maroon" title="c3">B</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">y</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Maroon" title="c3">B</font></span><br/>

<a NAME="E3:68"/><span class="lab"><font color="Green" title="E40">C</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E39">B</font></span></a>, <a class="ref" href="waybel_4.html#T57" title="WAYBEL_4:th.57">WAYBEL_4:57</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:68"/>
not <font color="Maroon" title="c5">y</font> <a href="waybel_4.html#R5" title="WAYBEL_4:pred.5">is_minimal_in</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">z</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="E6:68"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">and </span><br/><a NAME="E7:68"/><span class="lab"><font color="Green" title="E42">B1</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E38">A</font></span></a>, <a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E40">C</font></span></a>, <a class="ref" href="waybel_4.html#T57" title="WAYBEL_4:th.57">WAYBEL_4:57</a></span>;<br/>
<a NAME="E8:68"/><span class="lab"><font color="Green" title="E43">C1</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E42">B1</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">z9</font> = <font color="Maroon" title="c6">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="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>;<br/>
<a NAME="E10:68"/>
<font color="Maroon" title="c7">z9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E43">C1</font></span></a>, <a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E38">A</font></span></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></span>;<br/>
<a NAME="E11:68"/><span class="kw">then </span>
<font color="Maroon" title="c7">z9</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68"><span class="lab"><font color="Green" title="E42">B1</font></span></a>, <a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E38">A</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:68"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E39">B</font></span></a>, <a class="ref" href="waybel_4.html#T57" title="WAYBEL_4:th.57">WAYBEL_4:57</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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